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fricas
(1) -> <spad>
fricas
)abbrev package TYPEPKG TypePackage
TypePackage (T : Type) : Exports == Implementation where
  Exports == with
    typeof : T -> Type
  Implementation  == add
    typeof(x:T) == T
fricas
)abbrev package CMPTYPE CompareTypes
CompareTypes(T:Type,S:Type):Exports == Implementation  where
  SEX ==> SExpression
  Exports == with
    sameType? : (T, S) -> Boolean
  Implementation  == add
import TypePackage(T) import TypePackage(S)
sameType?(x:T,y:S):Boolean == test ( typeof(x)::SEX = typeof(y):: SEX )
fricas
)abbrev domain ROU RootOfUnity
++ Author: Kurt Pagani
++ Date Created: Fri Jun 01 17:24:19 CEST 2018
++ License: BSD
++ References: 
++   https://en.wikipedia.org/wiki/Root_of_unity
++   https://en.wikipedia.org/wiki/Principal_root_of_unity          
++ Description:
++   The nth roots of unity are, by definition, the roots of the polynomial 
++   $P(z)=z^n−1$, and are therefore algebraic numbers. As the polynomial $P$ 
++   is not irreducible - unless $n=1$, the primitive nth roots of unity are 
++   roots of an irreducible polynomial of lower degree, called the cyclotomic 
++   polynomial.
++
++ Group of nth roots of unity
++   The product and the multiplicative inverse of two nth roots of unity 
++   are also nth roots of unity. Therefore, the nth roots of unity form 
++   a group under multiplication.
++
++ Notes
++   Any algebraically closed field has exactly $n$ nth roots of unity if 
++   $n$ is not divisible by the characteristic of the field.
++
++   The significance of a root of unity being principal is that it is a 
++   necessary condition for the theory of the discrete Fourier transform 
++   to work out correctly.
++ 
++ Usage and Examples
++   X ==> Expression Complex Integer 
++   R ==> RootOfUnity(5,X)
++   z:X
++   r:=rootsOf(z^5-1) or zerosOf(z^5-1) or solve(z^5=1,'z) 
++   q:=[convert(t)$R for t in r]
++   [primitive?(t) for t in q]
++   [principal?(t) for t in q]
++
RootOfUnity(n,R) : Exports == Implementation where
n:PositiveInteger R:Ring
CTOF ==> CoercibleTo OutputForm
Exports == Join(Group,CTOF) with
convert : R -> % ++ Convert r:R to a n-th root of unity if r^n=1$R. retract : % -> R ++ Retract a n-th root of unity to a member of R. 1 : () -> % ++ The ring unit. primitive? : % -> Boolean ++ An nth root of unity is primitive if it is not a kth root of unity ++ for some smaller k. principal? : % -> Boolean ++ A principal n-th root of unity of a ring is an element a:R ++ satisfying the equations a^n=1$R, sum(a^(j*k),j=0..n-1)=0 ++ for all 1<=k<n. coerce : % -> OutputForm ++ Coerce to output form.
if R has ExpressionSpace then ExpressionSpace
Implementation == R add
Rep := R
convert(x) == x^n = 1$R => x error "Probably not a root of unity."
retract(x:%):R == x@Rep
primitive?(x:%):Boolean == b:List Boolean:=[test(x^m=1$R) for m in 1..n-1] not reduce(_and,b)
summ(a:R,m:PositiveInteger):R == s:List R:=[a^j for j in 0..m] reduce(_+,s)
principal?(x:%):Boolean == n=1 => false a:R:=x@Rep nn:PositiveInteger:=(n-1)::PositiveInteger b:List Boolean:=[test(summ(a^k,nn)=0$R) for k in 1..nn] reduce(_and,b)
fricas
)abbrev package DFT DiscreteFourierTransform
++ Author: Kurt Pagani
++ Date Created: Wed Mar 02 23:13:52 CET 2016
++ License: BSD
++ References: 
++   https://en.wikipedia.org/wiki/Discrete_Fourier_transform
++   https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general)
++
++ Description: 
++   Discrete Fourier transform (DFT) over any ring, commonly called
++   a number-theoretic transform (NTT) in the case of finite fields.  
++   Given a ring R and a principal n-th root of unity a, the package 
++   designator DFT(R,n,a) is used to compute various objects: 
++     dftMatrix()$DFT(R,n,a)
++   for instance, will compute and display a matrix M (the DFT matrix) 
++   such the discrete Fourier transform of the vector v is given by M*v.
++
++ Usage and Examples
++   X ==> EXPR COMPLEX INT
++   a:=convert(zerosOf(z^5-1).2)$ROU(5,X)
++   F ==> DFT(X,5,a)
++   m:=dftMatrix()$F
++   v:=vector [1::X,2,3,4,5]
++   w:=dft(v)$F
++   test(w=m*v)
++   test(idft(w)$F=v)
++   more? see test_dft.input
++
DiscreteFourierTransform(R,n,a) : Exports == Implementation where
R:Ring n:PositiveInteger a:RootOfUnity(n,R)
MATR ==> Matrix R
Exports == with
dftMatrix : () -> MATR ++ dftMatrix() computes and returns the DFT(R,n,a) matrix. dftInvMatrix : () -> MATR ++ dftInvMatrix() computes and returns the inverse DFT(R,n,a) matrix. dft : Vector R -> Vector R ++ dft(v) performs the DFT of the vector v. idft : Vector R -> Vector R ++ idft(v) performs the inverse DFT of the vector v.
Implementation == add
err1 := "The n-th root of unity provided is not principal." err2 := "Not invertible." err3 := "Wrong dimension of vector."
not principal?(a) => error err1
dftMatrix() == w:R:=retract(a) m:MATR:=matrix [[w^(i*j) for i in 0..n-1] for j in 0..n-1]
dftInvMatrix() == w:R:=retract(a) b:=recip(w) c:=recip(n*1::R) if b case R and c case R then m:MATR:=c * matrix [[b^(i*j) for i in 0..n-1] for j in 0..n-1] else error err2
dft(v) == #v ~= n => error err3 dftMatrix() * v
idft(v) == #v ~= n => error err3 dftInvMatrix() * v
fricas
)abbrev package UDFT UnitaryDiscreteFourierTransform
++ Description:
++   With unitary normalization constants 1/sqrt(n), the DFT becomes a 
++   unitary transformation, defined by a unitary matrix:
++     UDFT = DFT/sqrt(n)
++   Since IDFT = DFT*/N, we have IUDFT = sqrt(n) * IDFT.   
++   Moreover, |det(UDFT)|=1, IUDFT=UDFT*.
++
++ Usage and Examples
++   M:=udftMatrix()$UDFT(100)
++   IM:=udftInvMatrix()$UDFT(100)
++   w:=M*vector([i for i in 1..100])
++   IM*w
++   E:=M*IM -- needs some time!
++   trace(E) -- 100
++
++ Note that udft/iudft also compute udftMatrix internally each time
++ the functions are called, so it is certainly better to compute the
++ matrices once, if several transformations are to be performed.
++
UnitaryDiscreteFourierTransform(n) : Exports == Implementation where
n:PositiveInteger R ==> Expression Complex Integer DFT ==> DiscreteFourierTransform MATR ==> Matrix R
Exports == with
principalNthRootOfUnity : () -> RootOfUnity(n,R) ++ principalNthRootOfUnity() returns the principal n-th root ++ of unity which generates the cyclic group of the primitive ++ roots of unity as well as the Vandermonde matrix UDFT for ++ $\mathbb{C}^n$ ~ Expression Complex Integer. udftMatrix : () -> MATR ++ udftMatrix() computes and returns the UDFT(n) matrix. udftInvMatrix : () -> MATR ++ udftInvMatrix() computes and returns the inverse UDFT(n) matrix. udft : Vector R -> Vector R ++ udft(v) performs the UDFT of the vector v. iudft : Vector R -> Vector R ++ iudft(v) performs the inverse UDFT of the vector v.
Implementation == add
err1 := "Wrong dimension of vector."
principalNthRootOfUnity():RootOfUnity(n,R) == z:R:='z::R p:R:=z^n-1$R zop:List R:=zerosOf(p) a:R:=zop.2 convert(inv a)$RootOfUnity(n,R)
a:RootOfUnity(n,R):=principalNthRootOfUnity()
udftMatrix() == dftMatrix()$DFT(R,n,a) / sqrt(n::R)
udftInvMatrix() == dftInvMatrix()$DFT(R,n,a) * sqrt(n::R)
udft(v) == #v ~= n => error err1 udftMatrix() * v
iudft(v) == #v ~= n => error err1 udftInvMatrix() * v</spad>
fricas
Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6294884381899805856-25px001.spad
      using old system compiler.
   TYPEPKG abbreviates package TypePackage 
------------------------------------------------------------------------
   initializing NRLIB TYPEPKG for TypePackage 
   compiling into NRLIB TYPEPKG 
   compiling exported typeof : T$ -> Type
Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |TypePackage| REDEFINED
;;; *** |TypePackage| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor TypePackage Time: 0 seconds
finalizing NRLIB TYPEPKG Processing TypePackage for Browser database: --->-->TypePackage(constructor): Not documented!!!! --->-->TypePackage((typeof ((Type) T$))): Not documented!!!! --->-->TypePackage(): Missing Description ; compiling file "/var/aw/var/LatexWiki/TYPEPKG.NRLIB/TYPEPKG.lsp" (written 03 DEC 2024 07:19:37 PM):
; wrote /var/aw/var/LatexWiki/TYPEPKG.NRLIB/TYPEPKG.fasl ; compilation finished in 0:00:00.008 ------------------------------------------------------------------------ TypePackage is now explicitly exposed in frame initial TypePackage will be automatically loaded when needed from /var/aw/var/LatexWiki/TYPEPKG.NRLIB/TYPEPKG
CMPTYPE abbreviates package CompareTypes ------------------------------------------------------------------------ initializing NRLIB CMPTYPE for CompareTypes compiling into NRLIB CMPTYPE importing TypePackage T$ importing TypePackage S compiling exported sameType? : (T$,S) -> Boolean Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |CompareTypes| REDEFINED
;;; *** |CompareTypes| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor CompareTypes Time: 0 seconds
finalizing NRLIB CMPTYPE Processing CompareTypes for Browser database: --->-->CompareTypes(constructor): Not documented!!!! --->-->CompareTypes((sameType? ((Boolean) T$ S))): Not documented!!!! --->-->CompareTypes(): Missing Description ; compiling file "/var/aw/var/LatexWiki/CMPTYPE.NRLIB/CMPTYPE.lsp" (written 03 DEC 2024 07:19:37 PM):
; wrote /var/aw/var/LatexWiki/CMPTYPE.NRLIB/CMPTYPE.fasl ; compilation finished in 0:00:00.004 ------------------------------------------------------------------------ CompareTypes is now explicitly exposed in frame initial CompareTypes will be automatically loaded when needed from /var/aw/var/LatexWiki/CMPTYPE.NRLIB/CMPTYPE
ROU abbreviates domain RootOfUnity ------------------------------------------------------------------------ initializing NRLIB ROU for RootOfUnity compiling into NRLIB ROU compiling exported convert : R -> % Time: 0 SEC.
compiling exported retract : % -> R ROU;retract;%R;2 is replaced by x Time: 0 SEC.
compiling exported primitive? : % -> Boolean Time: 0.02 SEC.
compiling local summ : (R,PositiveInteger) -> R Time: 0 SEC.
compiling exported principal? : % -> Boolean Time: 0 SEC.
****** Domain: % already in scope augmenting %: (RetractableTo (Integer)) ****** Domain: R already in scope augmenting R: (ExpressionSpace) ****** Domain: % already in scope augmenting %: (Ring) ****** Domain: R already in scope augmenting R: (ExpressionSpace) ****** Domain: R already in scope augmenting R: (ExpressionSpace) (time taken in buildFunctor: 1316)
;;; *** |RootOfUnity| REDEFINED
;;; *** |RootOfUnity| REDEFINED Time: 0 SEC.
Warnings: [1] not known that (Comparable) is of mode (CATEGORY domain (SIGNATURE convert (% R)) (SIGNATURE retract (R %)) (SIGNATURE (One) (%)) (SIGNATURE primitive? ((Boolean) %)) (SIGNATURE principal? ((Boolean) %)) (SIGNATURE coerce ((OutputForm) %)) (IF (has R (ExpressionSpace)) (ATTRIBUTE (ExpressionSpace)) noBranch))
Cumulative Statistics for Constructor RootOfUnity Time: 0.03 seconds
finalizing NRLIB ROU Processing RootOfUnity for Browser database: --------constructor--------- --------(convert (% R))--------- --->-->RootOfUnity((convert (% R))): Improper first word in comments: Convert "Convert \\spad{r:R} to a \\spad{n}-th root of unity if \\spad{r^n=1}\\$\\spad{R}." --------(retract (R %))--------- --->-->RootOfUnity((retract (R %))): Improper first word in comments: Retract "Retract a \\spad{n}-th root of unity to a member of \\spad{R}." --------((One) (%))--------- --->-->RootOfUnity(((One) (%))): Improper first word in comments: The "The ring unit." --------(primitive? ((Boolean) %))--------- --->-->RootOfUnity((primitive? ((Boolean) %))): Improper first word in comments: An "An \\spad{n}th root of unity is primitive if it is not a \\spad{k}th root of unity for some smaller \\spad{k}." --------(principal? ((Boolean) %))--------- --->-->RootOfUnity((principal? ((Boolean) %))): Improper first word in comments: A "A principal \\spad{n}-th root of unity of a ring is an element a:R satisfying the equations \\spad{a^n=1}\\$\\spad{R},{} sum(a^(\\spad{j*k}),{}\\spad{j=0}..\\spad{n}-1)\\spad{=0} for all 1<=k<n." --------(coerce ((OutputForm) %))--------- --->-->RootOfUnity((coerce ((OutputForm) %))): Improper first word in comments: Coerce "Coerce to output form." ; compiling file "/var/aw/var/LatexWiki/ROU.NRLIB/ROU.lsp" (written 03 DEC 2024 07:19:37 PM):
; wrote /var/aw/var/LatexWiki/ROU.NRLIB/ROU.fasl ; compilation finished in 0:00:00.012 ------------------------------------------------------------------------ RootOfUnity is now explicitly exposed in frame initial RootOfUnity will be automatically loaded when needed from /var/aw/var/LatexWiki/ROU.NRLIB/ROU
DFT abbreviates package DiscreteFourierTransform ------------------------------------------------------------------------ initializing NRLIB DFT for DiscreteFourierTransform compiling into NRLIB DFT compiling exported dftMatrix : () -> Matrix R Time: 0 SEC.
compiling exported dftInvMatrix : () -> Matrix R Time: 0 SEC.
compiling exported dft : Vector R -> Vector R Time: 0 SEC.
compiling exported idft : Vector R -> Vector R Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |DiscreteFourierTransform| REDEFINED
;;; *** |DiscreteFourierTransform| REDEFINED Time: 0 SEC.
Warnings: [1] unknown Functor code (error (QREFELT % 9))
Cumulative Statistics for Constructor DiscreteFourierTransform Time: 0.02 seconds
finalizing NRLIB DFT Processing DiscreteFourierTransform for Browser database: --------constructor--------- --------(dftMatrix ((Matrix R)))--------- --------(dftInvMatrix ((Matrix R)))--------- --------(dft ((Vector R) (Vector R)))--------- --------(idft ((Vector R) (Vector R)))--------- ; compiling file "/var/aw/var/LatexWiki/DFT.NRLIB/DFT.lsp" (written 03 DEC 2024 07:19:37 PM):
; wrote /var/aw/var/LatexWiki/DFT.NRLIB/DFT.fasl ; compilation finished in 0:00:00.020 ------------------------------------------------------------------------ DiscreteFourierTransform is now explicitly exposed in frame initial DiscreteFourierTransform will be automatically loaded when needed from /var/aw/var/LatexWiki/DFT.NRLIB/DFT
UDFT abbreviates package UnitaryDiscreteFourierTransform ------------------------------------------------------------------------ initializing NRLIB UDFT for UnitaryDiscreteFourierTransform compiling into NRLIB UDFT Local variable err1 type redefined: (Wrong dimension of vector.) to (The n-th root of unity provided is not principal.) compiling exported principalNthRootOfUnity : () -> RootOfUnity(n,Expression Complex Integer) Time: 0.02 SEC.
compiling exported udftMatrix : () -> Matrix Expression Complex Integer Time: 0.01 SEC.
compiling exported udftInvMatrix : () -> Matrix Expression Complex Integer Time: 0 SEC.
compiling exported udft : Vector Expression Complex Integer -> Vector Expression Complex Integer Time: 0 SEC.
compiling exported iudft : Vector Expression Complex Integer -> Vector Expression Complex Integer Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |UnitaryDiscreteFourierTransform| REDEFINED
;;; *** |UnitaryDiscreteFourierTransform| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor UnitaryDiscreteFourierTransform Time: 0.06 seconds
finalizing NRLIB UDFT Processing UnitaryDiscreteFourierTransform for Browser database: --------constructor--------- --------(principalNthRootOfUnity ((RootOfUnity n (Expression (Complex (Integer))))))--------- --->-->UnitaryDiscreteFourierTransform((principalNthRootOfUnity ((RootOfUnity n (Expression (Complex (Integer))))))): Unexpected HT command: \mathbb "\\spad{principalNthRootOfUnity()} returns the principal \\spad{n}-th root of unity which generates the cyclic group of the primitive roots of unity as well as the Vandermonde matrix UDFT for \\$\\mathbb{\\spad{C}}\\spad{^n}\\$ ~ Expression Complex Integer." --------(udftMatrix ((Matrix (Expression (Complex (Integer))))))--------- --------(udftInvMatrix ((Matrix (Expression (Complex (Integer))))))--------- --------(udft ((Vector (Expression (Complex (Integer)))) (Vector (Expression (Complex (Integer))))))--------- --------(iudft ((Vector (Expression (Complex (Integer)))) (Vector (Expression (Complex (Integer))))))--------- ; compiling file "/var/aw/var/LatexWiki/UDFT.NRLIB/UDFT.lsp" (written 03 DEC 2024 07:19:37 PM):
; wrote /var/aw/var/LatexWiki/UDFT.NRLIB/UDFT.fasl ; compilation finished in 0:00:00.008 ------------------------------------------------------------------------ UnitaryDiscreteFourierTransform is now explicitly exposed in frame initial UnitaryDiscreteFourierTransform will be automatically loaded when needed from /var/aw/var/LatexWiki/UDFT.NRLIB/UDFT

Test flavours

fricas
--)co DFT
-- https://en.wikipedia.org/w/index.php?title= -- Discrete_Fourier_transform&action=edit&section=3 -- X ==> EXPR COMPLEX INT
Type: Void
fricas
a:=convert(zerosOf(z^4-1).2)$ROU(4,X)

\label{eq1}i(1)
Type: RootOfUnity?(4,Expression(Complex(Integer)))
fricas
F ==> DFT(X,4,inv a)  -- Note: inv(a) gives another result than a !!!
Type: Void
fricas
m:=dftMatrix()$F

\label{eq2}\left[ 
\begin{array}{cccc}
1 & 1 & 1 & 1 
\
1 & - i & - 1 & i 
\
1 & - 1 & 1 & - 1 
\
1 & i & - 1 & - i 
(2)
Type: Matrix(Expression(Complex(Integer)))
fricas
v:=vector [1::X,2-%i,-%i,-1+2*%i]

\label{eq3}\left[ 1, \:{2 - i}, \: - i , \:{- 1 +{2 \  i}}\right](3)
Type: Vector(Expression(Complex(Integer)))
fricas
w:=dft(v)$F

\label{eq4}\left[ 2, \:{- 2 -{2 \  i}}, \: -{2 \  i}, \:{4 +{4 \  i}}\right](4)
Type: Vector(Expression(Complex(Integer)))
fricas
test(w=m*v)

\label{eq5} \mbox{\rm true} (5)
Type: Boolean
fricas
test(idft(w)$F=v)

\label{eq6} \mbox{\rm true} (6)
Type: Boolean
fricas
determinant(m/2)

\label{eq7}i(7)
Type: Expression(Complex(Integer))
fricas
-- Examples
-----------
R:=IntegerMod 5

\label{eq8}\hbox{\axiomType{IntegerMod}\ } \left({5}\right)(8)
Type: Type
fricas
a:=2::R

\label{eq9}2(9)
Type: IntegerMod?(5)
fricas
n:=4

\label{eq10}4(10)
Type: PositiveInteger?
fricas
DFTZ5==>DFT(R,n,a)
Type: Void
fricas
dftMatrix()$DFTZ5

\label{eq11}\left[ 
\begin{array}{cccc}
1 & 1 & 1 & 1 
\
1 & 2 & 4 & 3 
\
1 & 4 & 1 & 4 
\
1 & 3 & 4 & 2 
(11)
Type: Matrix(IntegerMod?(5))
fricas
dftInvMatrix()$DFTZ5

\label{eq12}\left[ 
\begin{array}{cccc}
4 & 4 & 4 & 4 
\
4 & 2 & 1 & 3 
\
4 & 1 & 4 & 1 
\
4 & 3 & 1 & 2 
(12)
Type: Matrix(IntegerMod?(5))
fricas
dft([1,2,3,4])$DFTZ5

\label{eq13}\left[ 0, \: 4, \: 3, \: 2 \right](13)
Type: Vector(IntegerMod?(5))
fricas
idft([0,4,3,2])$DFTZ5

\label{eq14}\left[ 1, \: 2, \: 3, \: 4 \right](14)
Type: Vector(IntegerMod?(5))
fricas
-- R:=Expression Complex Integer
-- n:=3   
-- a:=exp(-2*%i*%pi/n)
-- ... no!
M:=udftMatrix()$UDFT(5)

\label{eq15}\left[ 
\begin{array}{ccccc}
{\frac{1}{\sqrt{5}}}&{\frac{1}{\sqrt{5}}}&{\frac{1}{\sqrt{5}}}&{\frac{1}{\sqrt{5}}}&{\frac{1}{\sqrt{5}}}
\
{\frac{1}{\sqrt{5}}}&{\frac{1}{{\sqrt{5}}\  \%z 3}}&{\frac{1}{{\sqrt{5}}\ {{\%z 3}^{2}}}}&{\frac{1}{{\sqrt{5}}\ {{\%z 3}^{3}}}}& -{\frac{1}{{{\sqrt{5}}\ {{\%z 3}^{3}}}+{{\sqrt{5}}\ {{\%z 3}^{2}}}+{{\sqrt{5}}\  \%z 3}+{\sqrt{5}}}}
\
{\frac{1}{\sqrt{5}}}&{\frac{1}{{\sqrt{5}}\ {{\%z 3}^{2}}}}& -{\frac{1}{{{\sqrt{5}}\ {{\%z 3}^{3}}}+{{\sqrt{5}}\ {{\%z 3}^{2}}}+{{\sqrt{5}}\  \%z 3}+{\sqrt{5}}}}&{\frac{1}{{\sqrt{5}}\  \%z 3}}&{\frac{1}{{\sqrt{5}}\ {{\%z 3}^{3}}}}
\
{\frac{1}{\sqrt{5}}}&{\frac{1}{{\sqrt{5}}\ {{\%z 3}^{3}}}}&{\frac{1}{{\sqrt{5}}\  \%z 3}}& -{\frac{1}{{{\sqrt{5}}\ {{\%z 3}^{3}}}+{{\sqrt{5}}\ {{\%z 3}^{2}}}+{{\sqrt{5}}\  \%z 3}+{\sqrt{5}}}}&{\frac{1}{{\sqrt{5}}\ {{\%z 3}^{2}}}}
\
{\frac{1}{\sqrt{5}}}& -{\frac{1}{{{\sqrt{5}}\ {{\%z 3}^{3}}}+{{\sqrt{5}}\ {{\%z 3}^{2}}}+{{\sqrt{5}}\  \%z 3}+{\sqrt{5}}}}&{\frac{1}{{\sqrt{5}}\ {{\%z 3}^{3}}}}&{\frac{1}{{\sqrt{5}}\ {{\%z 3}^{2}}}}&{\frac{1}{{\sqrt{5}}\  \%z 3}}
(15)
Type: Matrix(Expression(Complex(Integer)))
fricas
d:=determinant(M)

\label{eq16}\frac{{4 \ {{\%z 3}^{2}}}-{3 \  \%z 3}+ 4}{{2 \ {\sqrt{5}}\ {{\%z 3}^{2}}}-{{\sqrt{5}}\  \%z 3}+{2 \ {\sqrt{5}}}}(16)
Type: Expression(Complex(Integer))
fricas
test(d=1) -- test false does not necessarily mean d<>1

\label{eq17} \mbox{\rm false} (17)
Type: Boolean
fricas
test(d^2=1) -- true

\label{eq18} \mbox{\rm true} (18)
Type: Boolean
fricas
r:=retract(d) --
Type: AlgebraicNumber?

\label{eq19}\frac{{2 \ {\sqrt{5}}\ {{\%z 3}^{3}}}+{2 \ {\sqrt{5}}\ {{\%z 3}^{2}}}+{\sqrt{5}}}{5}(19)
Type: AlgebraicNumber?
fricas
test(r=1) --  true :)

\label{eq20} \mbox{\rm false} (20)
Type: Boolean
fricas
)set mess time on
N:=20 -- 20 is quick, 50 ~12sec, 100 ?long

\label{eq21}20(21)
Type: PositiveInteger?
fricas
Time: 0 sec
M:=udftMatrix()$UDFT(N)

\label{eq22}\left[ 
\begin{array}{cccccccccccccccccccc}
{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}& -{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}
\
{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}
\
{\frac{1}{2 \ {\sqrt{5}}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}& -{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}& -{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}& -{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}& -{\frac{1}{2 \ {\sqrt{5}}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}-{2 \ {\sqrt{5}}\  \%z 5}}}&{\frac{1}{{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}-{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}+{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}-{2 \ {\sqrt{5}}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{7}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{6}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{5}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{4}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{3}}}}&{\frac{1}{2 \ {\sqrt{5}}\ {{\%z 5}^{2}}}}&{\frac{1}{2 \ {\sqrt{5}}\  \%z 5}}
(22)
Type: Matrix(Expression(Complex(Integer)))
fricas
Time: 0.08 (EV) + 0.01 (OT) = 0.10 sec
IM:=udftInvMatrix()$UDFT(N)

\label{eq23}\left[ 
\begin{array}{cccccccccccccccccccc}
{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{10}}
\
{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\  \%z 5}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{7}}}-{{\sqrt{5}}\ {{\%z 5}^{5}}}+{{\sqrt{5}}\ {{\%z 5}^{3}}}-{{\sqrt{5}}\  \%z 5}}{10}}& -{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\  \%z 5}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{7}}}+{{\sqrt{5}}\ {{\%z 5}^{5}}}-{{\sqrt{5}}\ {{\%z 5}^{3}}}+{{\sqrt{5}}\  \%z 5}}{10}}
\
{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}
\
{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{7}}}-{{\sqrt{5}}\ {{\%z 5}^{5}}}+{{\sqrt{5}}\ {{\%z 5}^{3}}}-{{\sqrt{5}}\  \%z 5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\  \%z 5}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}& -{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{7}}}+{{\sqrt{5}}\ {{\%z 5}^{5}}}-{{\sqrt{5}}\ {{\%z 5}^{3}}}+{{\sqrt{5}}\  \%z 5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\  \%z 5}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}
\
{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}
\
{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}
\
{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{1
0}}& -{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}
\
{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{\sqrt{5}}\  \%z 5}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{7}}}-{{\sqrt{5}}\ {{\%z 5}^{5}}}+{{\sqrt{5}}\ {{\%z 5}^{3}}}-{{\sqrt{5}}\  \%z 5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}& -{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\  \%z 5}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{7}}}+{{\sqrt{5}}\ {{\%z 5}^{5}}}-{{\sqrt{5}}\ {{\%z 5}^{3}}}+{{\sqrt{5}}\  \%z 5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}
\
{\frac{\sqrt{5}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{\sqrt{5}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{\sqrt{5}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}
\
{\frac{\sqrt{5}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{7}}}-{{\sqrt{5}}\ {{\%z 5}^{5}}}+{{\sqrt{5}}\ {{\%z 5}^{3}}}-{{\sqrt{5}}\  \%z 5}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\  \%z 5}{10}}& -{\frac{\sqrt{5}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{7}}}+{{\sqrt{5}}\ {{\%z 5}^{5}}}-{{\sqrt{5}}\ {{\%z 5}^{3}}}+{{\sqrt{5}}\  \%z 5}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\  \%z 5}{10}}
\
{\frac{\sqrt{5}}{10}}& -{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{1
0}}& -{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{10}}& -{\frac{\sqrt{5}}{1
0}}&{\frac{\sqrt{5}}{10}}& -{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{1
0}}& -{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{10}}& -{\frac{\sqrt{5}}{1
0}}&{\frac{\sqrt{5}}{10}}& -{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{1
0}}& -{\frac{\sqrt{5}}{10}}&{\frac{\sqrt{5}}{10}}& -{\frac{\sqrt{5}}{1
0}}&{\frac{\sqrt{5}}{10}}& -{\frac{\sqrt{5}}{10}}
\
{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\  \%z 5}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{7}}}+{{\sqrt{5}}\ {{\%z 5}^{5}}}-{{\sqrt{5}}\ {{\%z 5}^{3}}}+{{\sqrt{5}}\  \%z 5}}{10}}& -{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\  \%z 5}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{7}}}-{{\sqrt{5}}\ {{\%z 5}^{5}}}+{{\sqrt{5}}\ {{\%z 5}^{3}}}-{{\sqrt{5}}\  \%z 5}}{10}}
\
{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}
\
{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{7}}}+{{\sqrt{5}}\ {{\%z 5}^{5}}}-{{\sqrt{5}}\ {{\%z 5}^{3}}}+{{\sqrt{5}}\  \%z 5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\  \%z 5}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}& -{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{7}}}-{{\sqrt{5}}\ {{\%z 5}^{5}}}+{{\sqrt{5}}\ {{\%z 5}^{3}}}-{{\sqrt{5}}\  \%z 5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\  \%z 5}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}
\
{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}
\
{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{\sqrt{5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{\sqrt{5}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{\sqrt{5}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{\sqrt{5}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{\sqrt{5}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}
\
{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{\sqrt{5}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}
\
{\frac{\sqrt{5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\  \%z 5}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{7}}}+{{\sqrt{5}}\ {{\%z 5}^{5}}}-{{\sqrt{5}}\ {{\%z 5}^{3}}}+{{\sqrt{5}}\  \%z 5}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}& -{\frac{\sqrt{5}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{\sqrt{5}}\  \%z 5}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{7}}}-{{\sqrt{5}}\ {{\%z 5}^{5}}}+{{\sqrt{5}}\ {{\%z 5}^{3}}}-{{\sqrt{5}}\  \%z 5}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}
\
{\frac{\sqrt{5}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{\sqrt{5}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{\sqrt{5}}{1
0}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{\sqrt{5}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}
\
{\frac{\sqrt{5}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{7}}}+{{\sqrt{5}}\ {{\%z 5}^{5}}}-{{\sqrt{5}}\ {{\%z 5}^{3}}}+{{\sqrt{5}}\  \%z 5}}{10}}&{\frac{-{{\sqrt{5}}\ {{\%z 5}^{6}}}+{{\sqrt{5}}\ {{\%z 5}^{4}}}-{{\sqrt{5}}\ {{\%z 5}^{2}}}+{\sqrt{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{1
0}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}& -{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}& -{\frac{{\sqrt{5}}\  \%z 5}{10}}& -{\frac{\sqrt{5}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{7}}}-{{\sqrt{5}}\ {{\%z 5}^{5}}}+{{\sqrt{5}}\ {{\%z 5}^{3}}}-{{\sqrt{5}}\  \%z 5}}{10}}&{\frac{{{\sqrt{5}}\ {{\%z 5}^{6}}}-{{\sqrt{5}}\ {{\%z 5}^{4}}}+{{\sqrt{5}}\ {{\%z 5}^{2}}}-{\sqrt{5}}}{1
0}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{7}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{6}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{5}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{4}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{3}}}{10}}&{\frac{{\sqrt{5}}\ {{\%z 5}^{2}}}{10}}&{\frac{{\sqrt{5}}\  \%z 5}{10}}
(23)
Type: Matrix(Expression(Complex(Integer)))
fricas
Time: 0.04 (EV) + 0.01 (OT) = 0.05 sec
w:=M*vector([i for i in 1..N])

\label{eq24}\begin{array}{@{}l}
\displaystyle
\left[{\frac{105}{\sqrt{5}}}, \: \right.
\
\
\displaystyle
\left.{\frac{{{10}\ {{\%z 5}^{7}}}+{{10}\ {{\%z 5}^{6}}}+{{10}\ {{\%z 5}^{3}}}-{{10}\  \%z 5}-{10}}{{\sqrt{5}}\ {{\%z 5}^{2}}}}, \: \right.
\
\
\displaystyle
\left.{\frac{{{10}\ {{\%z 5}^{6}}}-{10}}{{\sqrt{5}}\ {{\%z 5}^{2}}}}, \: \right.
\
\
\displaystyle
\left.{\frac{-{{10}\ {{\%z 5}^{7}}}+{{10}\ {{\%z 5}^{5}}}-{{1
0}\ {{\%z 5}^{4}}}-{{10}\ {{\%z 5}^{3}}}+{{10}\ {{\%z 5}^{2}}}-{10}}{{\sqrt{5}}\ {{\%z 5}^{7}}}}, \right.
\
\
\displaystyle
\left.\:{\frac{-{4 \ {{\%z 5}^{6}}}-{2 \ {{\%z 5}^{4}}}-{2 \ {{\%z 5}^{2}}}- 4}{{\sqrt{5}}\ {{\%z 5}^{6}}}}, \: \right.
\
\
\displaystyle
\left.{\frac{-{5 \ {{\%z 5}^{5}}}- 5}{{\sqrt{5}}\ {{\%z 5}^{5}}}}, \: \right.
\
\
\displaystyle
\left.{\frac{-{{10}\ {{\%z 5}^{4}}}+{{10}\ {{\%z 5}^{2}}}-{10}}{{\sqrt{5}}\ {{\%z 5}^{4}}}}, \: \right.
\
\
\displaystyle
\left.{\frac{{{10}\ {{\%z 5}^{5}}}-{{10}\ {{\%z 5}^{3}}}-{{10}\ {{\%z 5}^{2}}}+{10}}{{\sqrt{5}}\ {{\%z 5}^{4}}}}, \: \right.
\
\
\displaystyle
\left.{\frac{-{2 \ {{\%z 5}^{6}}}-{6 \ {{\%z 5}^{4}}}+{4 \ {{\%z 5}^{2}}}- 2}{{\sqrt{5}}\ {{\%z 5}^{6}}}}, \: \right.
\
\
\displaystyle
\left.{\frac{-{{10}\ {{\%z 5}^{7}}}+{{10}\ {{\%z 5}^{5}}}-{{1
0}\ {{\%z 5}^{4}}}+{{10}\  \%z 5}-{10}}{{\sqrt{5}}\ {{\%z 5}^{7}}}}, \right.
\
\
\displaystyle
\left.\: -{\frac{5}{\sqrt{5}}}, \: \right.
\
\
\displaystyle
\left.{\frac{-{{10}\ {{\%z 5}^{7}}}+{{10}\ {{\%z 5}^{6}}}-{{1
0}\ {{\%z 5}^{3}}}+{{10}\  \%z 5}-{10}}{{\sqrt{5}}\ {{\%z 5}^{2}}}}, \right.
\
\
\displaystyle
\left.\:{\frac{-{2 \ {{\%z 5}^{6}}}-{6 \ {{\%z 5}^{4}}}+{4 \ {{\%z 5}^{2}}}- 2}{{\sqrt{5}}\ {{\%z 5}^{4}}}}, \: \right.
\
\
\displaystyle
\left.{\frac{-{{10}\ {{\%z 5}^{7}}}+{{10}\ {{\%z 5}^{5}}}+{{1
0}\ {{\%z 5}^{4}}}-{{10}\ {{\%z 5}^{3}}}-{{10}\ {{\%z 5}^{2}}}+{10}}{{\sqrt{5}}\ {{\%z 5}^{7}}}}, \right.
\
\
\displaystyle
\left.\:{\frac{-{{10}\ {{\%z 5}^{2}}}+{10}}{{\sqrt{5}}\ {{\%z 5}^{4}}}}, \:{\frac{-{5 \ {{\%z 5}^{5}}}+ 5}{{\sqrt{5}}\ {{\%z 5}^{5}}}}, \: \right.
\
\
\displaystyle
\left.{\frac{-{4 \ {{\%z 5}^{6}}}-{2 \ {{\%z 5}^{4}}}-{2 \ {{\%z 5}^{2}}}+ 6}{{\sqrt{5}}\ {{\%z 5}^{4}}}}, \: \right.
\
\
\displaystyle
\left.{\frac{-{{10}\ {{\%z 5}^{5}}}+{{10}\ {{\%z 5}^{3}}}-{{1
0}\ {{\%z 5}^{2}}}+{10}}{{\sqrt{5}}\ {{\%z 5}^{4}}}}, \: \right.
\
\
\displaystyle
\left.{\frac{-{{10}\ {{\%z 5}^{6}}}+{{10}\ {{\%z 5}^{4}}}+{10}}{{\sqrt{5}}\ {{\%z 5}^{6}}}}, \: \right.
\
\
\displaystyle
\left.{\frac{-{{10}\ {{\%z 5}^{7}}}+{{10}\ {{\%z 5}^{5}}}+{{1
0}\ {{\%z 5}^{4}}}+{{10}\  \%z 5}+{10}}{{\sqrt{5}}\ {{\%z 5}^{7}}}}\right] 
(24)
Type: Vector(Expression(Complex(Integer)))
fricas
Time: 0.01 (EV) = 0.02 sec
IM*w

\label{eq25}\left[ 1, \: 2, \: 3, \: 4, \: 5, \: 6, \: 7, \: 8, \: 9, \:{1
0}, \:{11}, \:{12}, \:{13}, \:{14}, \:{15}, \:{16}, \:{17}, \:{1
8}, \:{19}, \:{20}\right](25)
Type: Vector(Expression(Complex(Integer)))
fricas
Time: 0.02 (EV) = 0.02 sec
E:=M*IM -- needs some time!

\label{eq26}\left[ 
\begin{array}{cccccccccccccccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 
(26)
Type: Matrix(Expression(Complex(Integer)))
fricas
Time: 0.23 (EV) = 0.23 sec
trace(E) -- N

\label{eq27}20(27)
Type: PositiveInteger?
fricas
Time: 0 sec
--)synonym lispversion )lisp (lisp-implementation-version) map(x+->x::Complex Float,m)

\label{eq28}\left[ 
\begin{array}{cccc}
{1.0}&{1.0}&{1.0}&{1.0}
\
{1.0}& - i & -{1.0}& i 
\
{1.0}& -{1.0}&{1.0}& -{1.0}
\
{1.0}& i & -{1.0}& - i 
(28)
Type: Matrix(Complex(Float))
fricas
Time: 0 sec
fN:=complexNumeric(exp(-2*%pi*%i/N))

\label{eq29}{0.9510565162 \<u> 9515357212}-{{0.3090169943 \</u> 749474241}\  i}(29)
Type: Complex(Float)
fricas
Time: 0.02 sec
MF:=map(x+->subst(x,%z5=fN),M)

\label{eq30}\left[ 
\begin{array}{cccccccccccccccccccc}
{0.2236067977 \<u> 4997896964}&{0.2236067977 \</u> 4997896964}&{0.2
236067977 \<u> 4997896964}&{0.2236067977 \</u> 4997896964}&{0.2236
067977 \<u> 4997896964}&{0.2236067977 \</u> 4997896964}&{0.2236067
977 \<u> 4997896964}&{0.2236067977 \</u> 4997896964}&{0.2236067977 \<u> 4997896964}&{0.2236067977 \</u> 4997896964}&{0.2236067977 \<u> 4997896964}&{0.2236067977 \</u> 4997896964}&{0.2236067977 \<u> 499
7896964}&{0.2236067977 \</u> 4997896964}&{0.2236067977 \<u> 499789
6964}&{0.2236067977 \</u> 4997896964}&{0.2236067977 \<u> 499789696
4}&{0.2236067977 \</u> 4997896964}&{0.2236067977 \<u> 4997896964}&{0.2
236067977 \</u> 4997896964}
\
{0.2236067977 \<u> 4997896964}&{{0.2126627020 \</u> 8800998305}+{{0.0
690983005 \<u> 6250525758 \</u> 9}\  i}}&{{0.1809016994 \<u> 3749474
241}+{{0.1314327780 \</u> 2978340151}\  i}}&{{0.1314327780 \<u> 29
78340151}+{{0.1809016994 \</u> 3749474241}\  i}}&{{0.0690983005 \<u> 6250525759}+{{0.2126627020 \</u> 8800998305}\  i}}&{{0.378804
6498 \<u> 4852188066 E - 21}+{{0.2236067977 \</u> 4997896964}\  i}}&{-{0.0690983005 \<u> 6250525758 \</u> 9}+{{0.2126627020 \<u> 880099830
5}\  i}}&{-{0.1314327780 \</u> 2978340151}+{{0.1809016994 \<u> 374
9474241}\  i}}&{-{0.1809016994 \</u> 3749474241}+{{0.1314327780 \<u> 2978340151}\  i}}&{-{0.2126627020 \</u> 8800998304}+{{0.06909
83005 \<u> 6250525759}\  i}}& -{0.2236067977 \</u> 4997896964}&{-{0.2
126627020 \<u> 8800998305}-{{0.0690983005 \</u> 6250525758 \<u> 9}\  i}}&{-{0.1809016994 \</u> 3749474241}-{{0.1314327780 \<u> 29783401
51}\  i}}&{-{0.1314327780 \</u> 2978340151}-{{0.1809016994 \<u> 37
49474241}\  i}}&{-{0.0690983005 \</u> 6250525759}-{{0.2126627020 \<u> 8800998305}\  i}}&{-{0.3788046498 \</u> 4852188066 E - 21}-{{0.2
236067977 \<u> 4997896964}\  i}}&{{0.0690983005 \</u> 6250525758 \<u> 9}-{{0.2126627020 \</u> 8800998305}\  i}}&{{0.1314327780 \<u> 2978
340151}-{{0.1809016994 \</u> 3749474241}\  i}}&{{0.1809016994 \<u> 3749474241}-{{0.1314327780 \</u> 2978340151}\  i}}&{{0.212662702
0 \<u> 8800998304}-{{0.0690983005 \</u> 6250525759}\  i}}
\
{0.2236067977 \<u> 4997896964}&{{0.1809016994 \</u> 3749474241}+{{0.1
314327780 \<u> 2978340151}\  i}}&{{0.0690983005 \</u> 6250525759}+{{0.2126627020 \<u> 8800998305}\  i}}&{-{0.0690983005 \</u> 625052
5758 \<u> 9}+{{0.2126627020 \</u> 8800998305}\  i}}&{-{0.180901699
4 \<u> 3749474241}+{{0.1314327780 \</u> 2978340151}\  i}}& -{0.223
6067977 \<u> 4997896964}&{-{0.1809016994 \</u> 3749474241}-{{0.131
4327780 \<u> 2978340151}\  i}}&{-{0.0690983005 \</u> 6250525759}-{{0.2
126627020 \<u> 8800998305}\  i}}&{{0.0690983005 \</u> 6250525758 \<u> 9}-{{0.2126627020 \</u> 8800998305}\  i}}&{{0.1809016994 \<u> 3749
474241}-{{0.1314327780 \</u> 2978340151}\  i}}&{0.2236067977 \<u> 4997896964}&{{0.1809016994 \</u> 3749474241}+{{0.1314327780 \<u> 2
978340151}\  i}}&{{0.0690983005 \</u> 6250525759}+{{0.2126627020 \<u> 8800998305}\  i}}&{-{0.0690983005 \</u> 6250525758 \<u> 9}+{{0.2
126627020 \</u> 8800998305}\  i}}&{-{0.1809016994 \<u> 3749474241}+{{0.1314327780 \</u> 2978340151}\  i}}& -{0.2236067977 \<u> 499789
6964}&{-{0.1809016994 \</u> 3749474241}-{{0.1314327780 \<u> 297834
0151}\  i}}&{-{0.0690983005 \</u> 6250525759}-{{0.2126627020 \<u> 8800998305}\  i}}&{{0.0690983005 \</u> 6250525758 \<u> 9}-{{0.2126
627020 \</u> 8800998305}\  i}}&{{0.1809016994 \<u> 3749474241}-{{0.1
314327780 \</u> 2978340151}\  i}}
\
{0.2236067977 \<u> 4997896964}&{{0.1314327780 \</u> 2978340151}+{{0.1
809016994 \<u> 3749474241}\  i}}&{-{0.0690983005 \</u> 6250525758 \<u> 9}+{{0.2126627020 \</u> 8800998305}\  i}}&{-{0.2126627020 \<u> 8800998304}+{{0.0690983005 \</u> 6250525759}\  i}}&{-{0.18090169
94 \<u> 3749474241}-{{0.1314327780 \</u> 2978340151}\  i}}&{-{0.37
88046498 \<u> 4852188066 E - 21}-{{0.2236067977 \</u> 4997896964}\  i}}&{{0.1809016994 \<u> 3749474241}-{{0.1314327780 \</u> 297834015
1}\  i}}&{{0.2126627020 \<u> 8800998305}+{{0.0690983005 \</u> 6250
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\
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4241}-{{0.1314327780 \<u> 2978340151}\  i}}&{{0.0690983005 \</u> 6
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\
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\
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\
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\
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\
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\
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36067977 \<u> 4997896964}& -{0.2236067977 \</u> 4997896964}&{0.223
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\
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\
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\
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\
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\
{0.2236067977 \<u> 4997896964}&{-{0.3788046498 \</u> 4852188066 E - 21}-{{0.2236067977 \<u> 4997896964}\  i}}& -{0.2236067977 \</u> 4997896964}&{{0.3788046498 \<u> 4852188066 E - 21}+{{0.22360679
77 \</u> 4997896964}\  i}}&{0.2236067977 \<u> 4997896964}&{-{0.378
8046498 \</u> 4852188066 E - 21}-{{0.2236067977 \<u> 4997896964}\  i}}& -{0.2236067977 \</u> 4997896964}&{{0.3788046498 \<u> 48521880
66 E - 21}+{{0.2236067977 \</u> 4997896964}\  i}}&{0.2236067977 \<u> 4997896964}&{-{0.3788046498 \</u> 4852188066 E - 21}-{{0.2236
067977 \<u> 4997896964}\  i}}& -{0.2236067977 \</u> 4997896964}&{{0.3
788046498 \<u> 4852188066 E - 21}+{{0.2236067977 \</u> 4997896964}\  i}}&{0.2236067977 \<u> 4997896964}&{-{0.3788046498 \</u> 485218806
6 E - 21}-{{0.2236067977 \<u> 4997896964}\  i}}& -{0.2236067977 \</u> 4997896964}&{{0.3788046498 \<u> 4852188066 E - 21}+{{0.22360
67977 \</u> 4997896964}\  i}}&{0.2236067977 \<u> 4997896964}&{-{0.3
788046498 \</u> 4852188066 E - 21}-{{0.2236067977 \<u> 4997896964}\  i}}& -{0.2236067977 \</u> 4997896964}&{{0.3788046498 \<u> 48521880
66 E - 21}+{{0.2236067977 \</u> 4997896964}\  i}}
\
{0.2236067977 \<u> 4997896964}&{{0.0690983005 \</u> 6250525758 \<u> 9}-{{0.2126627020 \</u> 8800998305}\  i}}&{-{0.1809016994 \<u> 374
9474241}-{{0.1314327780 \</u> 2978340151}\  i}}&{-{0.1809016994 \<u> 3749474241}+{{0.1314327780 \</u> 2978340151}\  i}}&{{0.069098
3005 \<u> 6250525759}+{{0.2126627020 \</u> 8800998305}\  i}}&{0.22
36067977 \<u> 4997896964}&{{0.0690983005 \</u> 6250525758 \<u> 9}-{{0.2
126627020 \</u> 8800998305}\  i}}&{-{0.1809016994 \<u> 3749474241}-{{0.1314327780 \</u> 2978340151}\  i}}&{-{0.1809016994 \<u> 374947
4241}+{{0.1314327780 \</u> 2978340151}\  i}}&{{0.0690983005 \<u> 6
250525759}+{{0.2126627020 \</u> 8800998305}\  i}}&{0.2236067977 \<u> 4997896964}&{{0.0690983005 \</u> 6250525758 \<u> 9}-{{0.2126627
020 \</u> 8800998305}\  i}}&{-{0.1809016994 \<u> 3749474241}-{{0.1
314327780 \</u> 2978340151}\  i}}&{-{0.1809016994 \<u> 3749474241}+{{0.1314327780 \</u> 2978340151}\  i}}&{{0.0690983005 \<u> 6250525
759}+{{0.2126627020 \</u> 8800998305}\  i}}&{0.2236067977 \<u> 499
7896964}&{{0.0690983005 \</u> 6250525758 \<u> 9}-{{0.2126627020 \</u> 8800998305}\  i}}&{-{0.1809016994 \<u> 3749474241}-{{0.13143277
80 \</u> 2978340151}\  i}}&{-{0.1809016994 \<u> 3749474241}+{{0.13
14327780 \</u> 2978340151}\  i}}&{{0.0690983005 \<u> 6250525759}+{{0.2
126627020 \</u> 8800998305}\  i}}
\
{0.2236067977 \<u> 4997896964}&{{0.1314327780 \</u> 2978340151}-{{0.1
809016994 \<u> 3749474241}\  i}}&{-{0.0690983005 \</u> 6250525759}-{{0.2126627020 \<u> 8800998305}\  i}}&{-{0.2126627020 \</u> 880099
8305}-{{0.0690983005 \<u> 6250525758 \</u> 9}\  i}}&{-{0.180901699
4 \<u> 3749474241}+{{0.1314327780 \</u> 2978340151}\  i}}&{{0.3788
046498 \<u> 4852188066 E - 21}+{{0.2236067977 \</u> 4997896964}\  i}}&{{0.1809016994 \<u> 3749474241}+{{0.1314327780 \</u> 297834015
1}\  i}}&{{0.2126627020 \<u> 8800998304}-{{0.0690983005 \</u> 6250
525759}\  i}}&{{0.0690983005 \<u> 6250525758 \</u> 9}-{{0.21266270
20 \<u> 8800998305}\  i}}&{-{0.1314327780 \</u> 2978340151}-{{0.18
09016994 \<u> 3749474241}\  i}}& -{0.2236067977 \</u> 4997896964}&{-{0.1314327780 \<u> 2978340151}+{{0.1809016994 \</u> 3749474241}\  i}}&{{0.0690983005 \<u> 6250525759}+{{0.2126627020 \</u> 880099830
5}\  i}}&{{0.2126627020 \<u> 8800998305}+{{0.0690983005 \</u> 6250
525758 \<u> 9}\  i}}&{{0.1809016994 \</u> 3749474241}-{{0.13143277
80 \<u> 2978340151}\  i}}&{-{0.3788046498 \</u> 4852188066 E - 21}-{{0.2236067977 \<u> 4997896964}\  i}}&{-{0.1809016994 \</u> 374947
4241}-{{0.1314327780 \<u> 2978340151}\  i}}&{-{0.2126627020 \</u> 8800998304}+{{0.0690983005 \<u> 6250525759}\  i}}&{-{0.06909830
05 \</u> 6250525758 \<u> 9}+{{0.2126627020 \</u> 8800998305}\  i}}&{{0.1
314327780 \<u> 2978340151}+{{0.1809016994 \</u> 3749474241}\  i}}
\
{0.2236067977 \<u> 4997896964}&{{0.1809016994 \</u> 3749474241}-{{0.1
314327780 \<u> 2978340151}\  i}}&{{0.0690983005 \</u> 6250525758 \<u> 9}-{{0.2126627020 \</u> 8800998305}\  i}}&{-{0.0690983005 \<u> 625
0525759}-{{0.2126627020 \</u> 8800998305}\  i}}&{-{0.1809016994 \<u> 3749474241}-{{0.1314327780 \</u> 2978340151}\  i}}& -{0.22360
67977 \<u> 4997896964}&{-{0.1809016994 \</u> 3749474241}+{{0.13143
27780 \<u> 2978340151}\  i}}&{-{0.0690983005 \</u> 6250525758 \<u> 9}+{{0.2126627020 \</u> 8800998305}\  i}}&{{0.0690983005 \<u> 6250525
759}+{{0.2126627020 \</u> 8800998305}\  i}}&{{0.1809016994 \<u> 37
49474241}+{{0.1314327780 \</u> 2978340151}\  i}}&{0.2236067977 \<u> 4997896964}&{{0.1809016994 \</u> 3749474241}-{{0.1314327780 \<u> 2
978340151}\  i}}&{{0.0690983005 \</u> 6250525758 \<u> 9}-{{0.21266
27020 \</u> 8800998305}\  i}}&{-{0.0690983005 \<u> 6250525759}-{{0.2
126627020 \</u> 8800998305}\  i}}&{-{0.1809016994 \<u> 3749474241}-{{0.1314327780 \</u> 2978340151}\  i}}& -{0.2236067977 \<u> 499789
6964}&{-{0.1809016994 \</u> 3749474241}+{{0.1314327780 \<u> 297834
0151}\  i}}&{-{0.0690983005 \</u> 6250525758 \<u> 9}+{{0.212662702
0 \</u> 8800998305}\  i}}&{{0.0690983005 \<u> 6250525759}+{{0.2126
627020 \</u> 8800998305}\  i}}&{{0.1809016994 \<u> 3749474241}+{{0.1
314327780 \</u> 2978340151}\  i}}
\
{0.2236067977 \<u> 4997896964}&{{0.2126627020 \</u> 8800998304}-{{0.0
690983005 \<u> 6250525759}\  i}}&{{0.1809016994 \</u> 3749474241}-{{0.1314327780 \<u> 2978340151}\  i}}&{{0.1314327780 \</u> 2978340
151}-{{0.1809016994 \<u> 3749474241}\  i}}&{{0.0690983005 \</u> 62
50525758 \<u> 9}-{{0.2126627020 \</u> 8800998305}\  i}}&{-{0.37880
46498 \<u> 4852188066 E - 21}-{{0.2236067977 \</u> 4997896964}\  i}}&{-{0.0690983005 \<u> 6250525759}-{{0.2126627020 \</u> 8800998305}\  i}}&{-{0.1314327780 \<u> 2978340151}-{{0.1809016994 \</u> 37494742
41}\  i}}&{-{0.1809016994 \<u> 3749474241}-{{0.1314327780 \</u> 29
78340151}\  i}}&{-{0.2126627020 \<u> 8800998305}-{{0.0690983005 \</u> 6250525758 \<u> 9}\  i}}& -{0.2236067977 \</u> 4997896964}&{-{0.2
126627020 \<u> 8800998304}+{{0.0690983005 \</u> 6250525759}\  i}}&{-{0.1809016994 \<u> 3749474241}+{{0.1314327780 \</u> 2978340151}\  i}}&{-{0.1314327780 \<u> 2978340151}+{{0.1809016994 \</u> 37494742
41}\  i}}&{-{0.0690983005 \<u> 6250525758 \</u> 9}+{{0.2126627020 \<u> 8800998305}\  i}}&{{0.3788046498 \</u> 4852188066 E - 21}+{{0.2
236067977 \<u> 4997896964}\  i}}&{{0.0690983005 \</u> 6250525759}+{{0.2126627020 \<u> 8800998305}\  i}}&{{0.1314327780 \</u> 2978340
151}+{{0.1809016994 \<u> 3749474241}\  i}}&{{0.1809016994 \</u> 37
49474241}+{{0.1314327780 \<u> 2978340151}\  i}}&{{0.2126627020 \</u> 8800998305}+{{0.0690983005 \<u> 6250525758 \</u> 9}\  i}}
(30)
Type: Matrix(Expression(Complex(Float)))
fricas
Time: 0.16 (IN) + 0.03 (OT) = 0.20 sec
trace(MF) -- 1+i

\label{eq31}{1.0}+{{1.0}\  i}(31)
Type: Complex(Float)
fricas
Time: 0 sec
determinant(MF) -- -i

\label{eq32}{0.242 E - 16}-{{0.9999999999 \<u> 9999998768}\  i}(32)
Type: Expression(Complex(Float))
fricas
Time: 0 sec




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