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last edited 1 year ago by Bill Page |
Edit detail for LinearOperator revision 5 of 63
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Editor: Bill Page
Time: 2011/03/08 16:43:14 GMT-8 |
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| Note: documentation and tests | ||
added: Linear transformations (operators) over n-dimensional cartesian vector spaces oveer a commutative ring $K$. Members of this domain are morphisms $K^n \to K^m$. Products, co-products and composition (grafting) of morphisms is implemented. Operators are represented internally as tensors. Operator composition and products can be visualized by directed graphs (read from top to bottom) such as:: n = 3 inputs \ | / \ / / \ | \ | / \/ / \ / \ \|/ \ / / \ / \ / \ \/ / \ \ / / \ \ / / \ \ / / \ \ / / \ | m = 2 outputs Lines (edges) in the graph represent vectors, nodes represent operators. Horizontal juxtaposition represents product. Vertical juxtaposition represents composition. added: ++ incoming vector added: ++ output vector added: added: added: added: added: added: ((x:%)^(p:NonNegativeInteger)):% == q:=subtractIfCan(p,1) q case NonNegativeInteger => x^q * x 1$% added: (x:% ** p:NonNegativeInteger):% == q:=subtractIfCan(p,1) q case NonNegativeInteger => x**q + x 0$% added: added: added: changed: - id():% == per [1,1,1] id():% == per [1,1,kroneckerDelta()$T] added: added: added: added: added: Construction operators: input and output changed: -X1:L:=inp[script(x,[[i]]) for i in 1..2] -arity X1 -Y1:L:=out[script(y,[[],[i]]) for i in 1..2] -arity Y1 -Y11:L:=inp[Y1,Y1] -arity(Y11) -\end{axiom} - -\begin{axiom} -XX1:=X1*X1 -arity(XX1) -\end{axiom} -\begin{axiom} -YY1:=Y1*Y1 -arity(YY1) -\end{axiom} -\begin{axiom} -Y1X1:=Y1*X1 -arity(Y1X1) -\end{axiom} - -\begin{axiom} -XY11:=X1 Y1 -arity(XY11) -YX11:= Y1 X1 -arity(YX11) A1:L:=inp[script(a,[[1,i]]) for i in 1..2] arity A1 A2:L:=inp[script(a,[[2,i]]) for i in 1..2] A:L:=inp[A1,A2] arity A B1:L:=out[script(b,[[],[1,i]]) for i in 1..2] arity B1 B2:L:=out[script(b,[[],[2,i]]) for i in 1..2] B:L:=out[B1,B2] arity B \end{axiom} Powers \begin{axiom} A3p:=(A1*A1)*A1 arity(A3p) test(A3p=A1*(A1*A1)) test(A3p=A1^3) \end{axiom} \begin{axiom} B3p:=(B1*B1)*B1 arity(B3p) test(B3p=B1*(B1*B1)) test(B3p=B1^3) \end{axiom} Sums \begin{axiom} A3s:=(A1+A1)+A1 arity(A3s) test(A3s=A1*(A1*A1)) test(A3s=A1^3) \end{axiom} \begin{axiom} B3s:=(B1+B1)+B1 arity(B3s) test(B3s=B1*(B1*B1)) test(B3s=B1^3) \end{axiom} Expected error \begin{axiom} B1A1:=B1*A1 arity(B1A1) \end{axiom} Composition \begin{axiom} AB11:=A1 B1 arity(AB11) BA11:= B1 A1 arity(BA11) AB := A B arity(AB) BA := B A arity(BA) changed: -arity(X2) -XX21:=X2 X1 -arity(XX21) XB21:=X2 B1 arity(XB21) removed: -arity(Y2) added: \end{axiom} Multiple inputs and outputs \begin{axiom} changed: -WX1:=W X1 -arity(WX1) -\end{axiom} WB1:=W B1 arity(WB1) \end{axiom}
Linear transformations (operators) over n-dimensional cartesian vector spaces oveer a commutative ring 
. Members of this domain are morphisms 
. Products, co-products and composition (grafting) of morphisms is implemented. Operators are represented internally as tensors.
Operator composition and products can be visualized by directed graphs (read from top to bottom) such as:
n = 3 inputs
\ | / \ / / \ |
\ | / \/ / \ / \
\|/ \ / / \ / \
/ \ \/ / \ \ /
/ \ \ / / \ \ /
/ \ \ / / \ |
m = 2 outputs
Lines (edges) in the graph represent vectors, nodes represent operators. Horizontal juxtaposition represents product. Vertical juxtaposition represents composition.
spad
)abbrev domain LIN LinearOperator LinearOperator(dim:NonNegativeInteger,K:CommutativeRing): Join(Ring, BiModule(K, K)) with arity: % -> DirectProduct(2, NonNegativeInteger) elt: (%, %) -> % id: ()->% inp: List K -> % ++ incoming vector inp: List % -> % out: List K -> % ++ output vector out: List % -> % coerce: SquareMatrix(dim, K) -> % == add import List NonNegativeInteger T == CartesianTensor(1, dim, K) Rep == Record(n:NonNegativeInteger, m:NonNegativeInteger, t:T) rep(x:%):Rep == x pretend Rep per(x:Rep):% == x pretend %
arity(x:%):DirectProduct(2,NonNegativeInteger) == directProduct [rep(x).n, rep(x).m]
0 == per [0,0, 0]
(x:% + y:%):% == rep(x).t=0 => per [rep(y).n,rep(y).m, rep(y).t] rep(y).t=0 => per [rep(x).n, rep(x).m, rep(x).t] rep(x).n ~= rep(y).n or rep(x).m ~= rep(y).m => error "arity" per [rep(x).n, rep(x).m, rep(x).t+rep(y).t]
(x:% - y:%):% == rep(x).t=0 => per [rep(y).n,rep(y).m, -rep(y).t] rep(y).t=0 => per [rep(x).n, rep(x).m, rep(x).t] rep(x).n ~= rep(y).n or rep(x).m ~= rep(y).m => error "arity" per [rep(x).n, rep(x).m, rep(x).t-rep(y).t]
1 == per [0,0, 1]
(x:% * y:%):% == rep(x).n ~= rep(y).n => error "arity" r:=product(rep(x).t,rep(y).t) u:=1$DirectProduct(dim, K)::T ud:=product(u, kroneckerDelta()$T) for i in 1..rep(x).n repeat --output("rank", rank(r)::OutputForm)$OutputPackage --output("n", rep(x).n::OutputForm)$OutputPackage r:=contract(contract(ud, 1, r, rep(x).n+1), 1, rep(x).n+1) --output("rank", rank(r)::OutputForm)$OutputPackage per [rep(x).n, rep(x).m+rep(y).m, r] ((x:%)^(p:NonNegativeInteger)):% == q:=subtractIfCan(p, 1) q case NonNegativeInteger => x^q * x 1$%
(x:% + y:%):% == rep(x).m ~= rep(y).m => error "arity" r:=product(rep(x).t,rep(y).t) u:=1$DirectProduct(dim, K)::T du:=product(kroneckerDelta()$T, u) for i in 1..rep(y).m repeat --output("rank", rank(r)::OutputForm)$OutputPackage --output("m", rep(y).m::OutputForm)$OutputPackage r:=contract(contract(du, 1, r, rep(y).m+1), 1, rep(y).m+1) --output("rank", rank(r)::OutputForm)$OutputPackage per [rep(x).n+rep(y).m, rep(y).m, r]
(x:% ** p:NonNegativeInteger):% == q:=subtractIfCan(p,1) q case NonNegativeInteger => x**q + x 0$%
(x:% = y:%):Boolean == rep(x).n ~= rep(y).n or rep(x).m ~= rep(y).m => error "arity" rep(x).t = rep(y).t
(x:K * y:%):% == per [rep(y).n,rep(y).m, x*rep(y).t]
(x:% * y:K):% == per [rep(x).n,rep(x).m, rep(x).t*y]
elt(x:%,y:%):% == r:=product(rep(x).t, rep(y).t) yn:=rep(y).n -- outputs of y xm:=rep(x).m -- inputs of x while yn>0 and xm>0 repeat output("yn", yn::OutputForm)$OutputPackage output("xm", xm::OutputForm)$OutputPackage output("rank", rank(r)::OutputForm)$OutputPackage r:=contract(r, rep(y).n+xm, yn+rep(y).m) yn:=subtractIfCan(yn, 1)::NonNegativeInteger xm:=subtractIfCan(xm, 1)::NonNegativeInteger per [rep(x).n+xm, yn+rep(y).m, r]
id():% == per [1,1, kroneckerDelta()$T]
inp(x:List K):% == per [1,0, entries(x)::T]
inp(x:List %):% == #removeDuplicates([rep(y).n for y in x])~=1 or #removeDuplicates([rep(y).m for y in x])~=1 => error "arity" per [rep(first x).n+1,rep(first x).m, [rep(y).t for y in x]::T]$Rep
out(x:List K):% == per [0,1, entries(x)::T]
out(x:List %):% == #removeDuplicates([rep(y).n for y in x])~=1 or #removeDuplicates([rep(y).m for y in x])~=1 => error "arity" per [rep(first x).n,rep(first x).m+1, [rep(y).t for y in x]::T]$Rep
coerce(x:%):OutputForm == (rep(x).t)::OutputForm
spad
Compiling FriCAS source code from file
/var/zope2/var/LatexWiki/74143022208027506-25px001.spad using old
system compiler.
LIN abbreviates domain LinearOperator
------------------------------------------------------------------------
initializing NRLIB LIN for LinearOperator
compiling into NRLIB LIN
importing List NonNegativeInteger
compiling local rep : $ -> Record(n: NonNegativeInteger,
compiling local per : Record(n: NonNegativeInteger,
compiling exported arity : $ -> DirectProduct(2,
compiling exported Zero : () -> $
Time: 0.01 SEC.
compiling exported + : ($,
compiling exported - : ($,
compiling exported One : () -> $
Time: 0 SEC.
compiling exported * : ($,
compiling exported ^ : ($,
compiling exported + : ($,
compiling local ** : ($,
compiling exported = : ($,
compiling exported * : (K,
compiling exported * : ($,
compiling exported elt : ($,
compiling exported id : () -> $
Time: 0.01 SEC.
compiling exported inp : List K -> $
Time: 0.07 SEC.
compiling exported inp : List $ -> $
Time: 0.10 SEC.
compiling exported out : List K -> $
Time: 0.01 SEC.
compiling exported out : List $ -> $
Time: 0.07 SEC.
compiling exported coerce : $ -> OutputForm
Time: 0.01 SEC.
(time taken in buildFunctor: 10)
;;; *** |LinearOperator| REDEFINED
;;; *** |LinearOperator| REDEFINED
Time: 0.01 SEC.
Cumulative Statistics for Constructor LinearOperator
Time: 0.72 seconds
finalizing NRLIB LIN
Processing LinearOperator for Browser database:
--->-->LinearOperator((arity ((DirectProduct 2 (NonNegativeInteger)) %))): Not documented!!!!
--->-->LinearOperator((elt (% % %))): Not documented!!!!
--->-->LinearOperator((id (%))): Not documented!!!!
--------(inp (% (List K)))---------
--->-->LinearOperator((inp (% (List %)))): Not documented!!!!
--------(out (% (List K)))---------
--->-->LinearOperator((out (% (List %)))): Not documented!!!!
--->-->LinearOperator((coerce (% (SquareMatrix dim K)))): Not documented!!!!
--->-->LinearOperator(constructor): Not documented!!!!
--->-->LinearOperator(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/LIN.NRLIB/LIN.lsp" (written 08 MAR 2011 04:42:58 PM):
; compiling (/VERSIONCHECK 2)
; compiling (PUT (QUOTE |LIN;rep|) ...)
; compiling (DEFUN |LIN;rep| ...)
; compiling (PUT (QUOTE |LIN;per|) ...)
; compiling (DEFUN |LIN;per| ...)
; compiling (DEFUN |LIN;arity;$Dp;3| ...)
; compiling (DEFUN |LIN;Zero;$;4| ...)
; compiling (DEFUN |LIN;+;3$;5| ...)
; compiling (DEFUN |LIN;-;3$;6| ...)
; compiling (DEFUN |LIN;One;$;7| ...)
; compiling (DEFUN |LIN;*;3$;8| ...)
; compiling (DEFUN |LIN;^;$Nni$;9| ...)
; compiling (DEFUN |LIN;+;3$;10| ...)
; compiling (DEFUN |LIN;**| ...)
; compiling (DEFUN |LIN;=;2$B;12| ...)
; compiling (DEFUN |LIN;*;K2$;13| ...)
; compiling (DEFUN |LIN;*;$K$;14| ...)
; compiling (DEFUN |LIN;elt;3$;15| ...)
; compiling (DEFUN |LIN;id;$;16| ...)
; compiling (DEFUN |LIN;inp;L$;17| ...)
; compiling (DEFUN |LIN;inp;L$;18| ...)
; compiling (DEFUN |LIN;out;L$;19| ...)
; compiling (DEFUN |LIN;out;L$;20| ...)
; compiling (DEFUN |LIN;coerce;$Of;21| ...)
; compiling (DEFUN |LinearOperator| ...)
; compiling (DEFUN |LinearOperator;| ...)
; compiling (MAKEPROP (QUOTE |LinearOperator|) ...)
; /var/zope2/var/LatexWiki/LIN.NRLIB/LIN.fasl written
; compilation finished in 0:00:00.672
------------------------------------------------------------------------
LinearOperator is now explicitly exposed in frame initial
LinearOperator will be automatically loaded when needed from
/var/zope2/var/LatexWiki/LIN.NRLIB/LINConstruction operators: input and output
axiom
L:=LIN(2,FRAC POLY INT)
 | (1) |
Type: Domain
axiom
A1:L:=inp[script(a,[[1, i]]) for i in 1..2]
![\label{eq2}\left[{a_{1, \: 1}}, \:{a_{1, \: 2}}\right]](images/4321625372397848333-16.0px.png "\label{eq2}\left[{a_{1, \: 1}}, \:{a_{1, \: 2}}\right]") | (2) |
Type: LinearOperator?(2,
axiom
arity A1
![\label{eq3}\left[ 1, \: 0 \right]](images/3375084741431016225-16.0px.png "\label{eq3}\left[ 1, \: 0 \right]") | (3) |
axiom
A2:L:=inp[script(a,[[2, i]]) for i in 1..2]
![\label{eq4}\left[{a_{2, \: 1}}, \:{a_{2, \: 2}}\right]](images/724258529222624807-16.0px.png "\label{eq4}\left[{a_{2, \: 1}}, \:{a_{2, \: 2}}\right]") | (4) |
Type: LinearOperator?(2,
axiom
A:L:=inp[A1,A2]
![\label{eq5}\left[
\begin{array}{cc}
{a_{1, \: 1}}&{a_{1, \: 2}}
\
{a_{2, \: 1}}&{a_{2, \: 2}}](images/4924115446897861492-16.0px.png "\label{eq5}\left[
\begin{array}{cc}
{a_{1, \: 1}}&{a_{1, \: 2}}
\
{a_{2, \: 1}}&{a_{2, \: 2}}") | (5) |
Type: LinearOperator?(2,
axiom
arity A
![\label{eq6}\left[ 2, \: 0 \right]](images/7880863799722180819-16.0px.png "\label{eq6}\left[ 2, \: 0 \right]") | (6) |
axiom
B1:L:=out[script(b,[[], [1, i]]) for i in 1..2]
![\label{eq7}\left[{b^{1, \: 1}}, \:{b^{1, \: 2}}\right]](images/3204221659305033192-16.0px.png "\label{eq7}\left[{b^{1, \: 1}}, \:{b^{1, \: 2}}\right]") | (7) |
Type: LinearOperator?(2,
axiom
arity B1
![\label{eq8}\left[ 0, \: 1 \right]](images/5536053462494288574-16.0px.png "\label{eq8}\left[ 0, \: 1 \right]") | (8) |
axiom
B2:L:=out[script(b,[[], [2, i]]) for i in 1..2]
![\label{eq9}\left[{b^{2, \: 1}}, \:{b^{2, \: 2}}\right]](images/8417299754634676252-16.0px.png "\label{eq9}\left[{b^{2, \: 1}}, \:{b^{2, \: 2}}\right]") | (9) |
Type: LinearOperator?(2,
axiom
B:L:=out[B1,B2]
![\label{eq10}\left[
\begin{array}{cc}
{b^{1, \: 1}}&{b^{1, \: 2}}
\
{b^{2, \: 1}}&{b^{2, \: 2}}](images/8595235799847785773-16.0px.png "\label{eq10}\left[
\begin{array}{cc}
{b^{1, \: 1}}&{b^{1, \: 2}}
\
{b^{2, \: 1}}&{b^{2, \: 2}}") | (10) |
Type: LinearOperator?(2,
axiom
arity B
![\label{eq11}\left[ 0, \: 2 \right]](images/8078613850184813856-16.0px.png "\label{eq11}\left[ 0, \: 2 \right]") | (11) |
Powers
axiom
A3p:=(A1*A1)*A1
![\label{eq12}\begin{array}{@{}l}
\displaystyle
\left[{{4 \ {a_{1, \: 1}}\ {{a_{1, \: 2}}^2}}+{8 \ {{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}+{4 \ {{a_{1, \: 1}}^3}}}, \: \right.
\
\
\displaystyle
\left.{{4 \ {{a_{1, \: 2}}^3}}+{8 \ {a_{1, \: 1}}\ {{a_{1, \: 2}}^2}}+{4 \ {{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}}\right]](images/1117248889202897521-16.0px.png "\label{eq12}\begin{array}{@{}l}
\displaystyle
\left[{{4 \ {a_{1, \: 1}}\ {{a_{1, \: 2}}^2}}+{8 \ {{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}+{4 \ {{a_{1, \: 1}}^3}}}, \: \right.
\
\
\displaystyle
\left.{{4 \ {{a_{1, \: 2}}^3}}+{8 \ {a_{1, \: 1}}\ {{a_{1, \: 2}}^2}}+{4 \ {{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}}\right]") | (12) |
Type: LinearOperator?(2,
axiom
arity(A3p)
![\label{eq13}\left[ 1, \: 0 \right]](images/4319565934820351433-16.0px.png "\label{eq13}\left[ 1, \: 0 \right]") | (13) |
axiom
test(A3p=A1*(A1*A1))
 | (14) |
Type: Boolean
axiom
test(A3p=A1^3)
 | (15) |
Type: Boolean
axiom
B3p:=(B1*B1)*B1
![\label{eq16}\begin{array}{@{}l}
\displaystyle
\left[{\left[
\begin{array}{cc}
{{b^{1, \: 1}}^3}&{{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}
\
{{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}&{{b^{1, \: 1}}\ {{b^{1, \: 2}}^2}}](images/7102526790177766491-16.0px.png "\label{eq16}\begin{array}{@{}l}
\displaystyle
\left[{\left[
\begin{array}{cc}
{{b^{1, \: 1}}^3}&{{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}
\
{{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}&{{b^{1, \: 1}}\ {{b^{1, \: 2}}^2}}") | (16) |
Type: LinearOperator?(2,
axiom
arity(B3p)
![\label{eq17}\left[ 0, \: 3 \right]](images/3112017029395132859-16.0px.png "\label{eq17}\left[ 0, \: 3 \right]") | (17) |
axiom
test(B3p=B1*(B1*B1))
 | (18) |
Type: Boolean
axiom
test(B3p=B1^3)
 | (19) |
Type: Boolean
Sums
axiom
A3s:=(A1+A1)+A1
![\label{eq20}\begin{array}{@{}l}
\displaystyle
\left[{\left[
\begin{array}{cc}
{{a_{1, \: 1}}^3}&{{{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}
\
{{{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}&{{a_{1, \: 1}}\ {{a_{1, \: 2}}^2}}](images/3706160820046993834-16.0px.png "\label{eq20}\begin{array}{@{}l}
\displaystyle
\left[{\left[
\begin{array}{cc}
{{a_{1, \: 1}}^3}&{{{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}
\
{{{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}&{{a_{1, \: 1}}\ {{a_{1, \: 2}}^2}}") | (20) |
Type: LinearOperator?(2,
axiom
arity(A3s)
![\label{eq21}\left[ 1, \: 0 \right]](images/1637303886053562416-16.0px.png "\label{eq21}\left[ 1, \: 0 \right]") | (21) |
axiom
test(A3s=A1*(A1*A1))
 | (22) |
Type: Boolean
axiom
test(A3s=A1^3)
 | (23) |
Type: Boolean
axiom
B3s:=(B1+B1)+B1
![\label{eq24}\begin{array}{@{}l}
\displaystyle
\left[{{{b^{1, \: 1}}\ {{b^{1, \: 2}}^2}}+{2 \ {{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}+{{b^{1, \: 1}}^3}}, \: \right.
\
\
\displaystyle
\left.{{{b^{1, \: 2}}^3}+{2 \ {b^{1, \: 1}}\ {{b^{1, \: 2}}^2}}+{{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}}\right]](images/6477068065764311044-16.0px.png "\label{eq24}\begin{array}{@{}l}
\displaystyle
\left[{{{b^{1, \: 1}}\ {{b^{1, \: 2}}^2}}+{2 \ {{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}+{{b^{1, \: 1}}^3}}, \: \right.
\
\
\displaystyle
\left.{{{b^{1, \: 2}}^3}+{2 \ {b^{1, \: 1}}\ {{b^{1, \: 2}}^2}}+{{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}}\right]") | (24) |
Type: LinearOperator?(2,
axiom
arity(B3s)
![\label{eq25}\left[ 2, \: 1 \right]](images/7372521291154450594-16.0px.png "\label{eq25}\left[ 2, \: 1 \right]") | (25) |
axiom
test(B3s=B1*(B1*B1))
>> Error detected within library code: arity
Expected error
axiom
B1A1:=B1*A1
>> Error detected within library code: arity
Composition
axiom
AB11:=A1 B1
![\label{eq26}\left[
\begin{array}{cc}
{{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}}
\
{{b^{1, \: 1}}\ {a_{1, \: 2}}}&{{b^{1, \: 2}}\ {a_{1, \: 2}}}](images/1743540498809314582-16.0px.png "\label{eq26}\left[
\begin{array}{cc}
{{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}}
\
{{b^{1, \: 1}}\ {a_{1, \: 2}}}&{{b^{1, \: 2}}\ {a_{1, \: 2}}}") | (26) |
Type: LinearOperator?(2,
axiom
arity(AB11)
![\label{eq27}\left[ 1, \: 1 \right]](images/5952721708021568767-16.0px.png "\label{eq27}\left[ 1, \: 1 \right]") | (27) |
axiom
BA11:= B1 A1
yn 1 xm 1 rank 2
 | (28) |
Type: LinearOperator?(2,
axiom
arity(BA11)
![\label{eq29}\left[ 0, \: 0 \right]](images/4501844044754249791-16.0px.png "\label{eq29}\left[ 0, \: 0 \right]") | (29) |
axiom
AB := A B
![\label{eq30}\left[
\begin{array}{cc}
{\left[
\begin{array}{cc}
{{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}}
\
{{b^{2, \: 1}}\ {a_{1, \: 1}}}&{{b^{2, \: 2}}\ {a_{1, \: 1}}}](images/7848290865419487405-16.0px.png "\label{eq30}\left[
\begin{array}{cc}
{\left[
\begin{array}{cc}
{{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}}
\
{{b^{2, \: 1}}\ {a_{1, \: 1}}}&{{b^{2, \: 2}}\ {a_{1, \: 1}}}") | (30) |
Type: LinearOperator?(2,
axiom
arity(AB)
![\label{eq31}\left[ 2, \: 2 \right]](images/8007342481487348264-16.0px.png "\label{eq31}\left[ 2, \: 2 \right]") | (31) |
axiom
BA := B A
yn 2 xm 2 rank 4 yn 1 xm 1 rank 2
>> Error detected within library code: Improper index for contraction
Multiple inputs and outputs
axiom
W:L:=out[inp[inp([script(w,[[i, j], [k]]) for j in 1..2])$L for i in 1..2] for k in 1..2]
![\label{eq32}\begin{array}{@{}l}
\displaystyle
\left[{\left[
\begin{array}{cc}
{w_{1, \: 1}^{1}}&{w_{1, \: 2}^{1}}
\
{w_{2, \: 1}^{1}}&{w_{2, \: 2}^{1}}](images/4385751582784471522-16.0px.png "\label{eq32}\begin{array}{@{}l}
\displaystyle
\left[{\left[
\begin{array}{cc}
{w_{1, \: 1}^{1}}&{w_{1, \: 2}^{1}}
\
{w_{2, \: 1}^{1}}&{w_{2, \: 2}^{1}}") | (32) |
Type: LinearOperator?(2,
axiom
WB1:=W B1
![\label{eq33}\left[
\begin{array}{cc}
{\left[
\begin{array}{cc}
{{b^{1, \: 1}}\ {w_{1, \: 1}^{1}}}&{{b^{1, \: 2}}\ {w_{1, \: 1}^{1}}}
\
{{b^{1, \: 1}}\ {w_{1, \: 2}^{1}}}&{{b^{1, \: 2}}\ {w_{1, \: 2}^{1}}}](images/3509487409130179300-16.0px.png "\label{eq33}\left[
\begin{array}{cc}
{\left[
\begin{array}{cc}
{{b^{1, \: 1}}\ {w_{1, \: 1}^{1}}}&{{b^{1, \: 2}}\ {w_{1, \: 1}^{1}}}
\
{{b^{1, \: 1}}\ {w_{1, \: 2}^{1}}}&{{b^{1, \: 2}}\ {w_{1, \: 2}^{1}}}") | (33) |
Type: LinearOperator?(2,
axiom
arity(WB1)
![\label{eq34}\left[ 3, \: 1 \right]](images/3823837797590606063-16.0px.png "\label{eq34}\left[ 3, \: 1 \right]") | (34) |