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	FriCAS Algebra
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LinearOperator

last edited 2 years ago by Bill Page

Edit detail for LinearOperator revision 14 of 63

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Editor: page Time: 2011/04/09 22:14:38 GMT-7
Note: bug fix in product

changed:
-Monoidal(R:AbelianSemiGroup):Category == SemiRng with
)abbrev category MONAL Monoidal
Monoidal(R:AbelianSemiGroup):Category == Monoid with

changed:
-    elt:(%,%) -> %
-      ++ horizontal composition (product)
    apply:(%,%) -> %
      ++ horizontal composition
    coerce:(x:List None) -> %
      ++ identity for composition

changed:
-    (f:% / g:%):% == g*f
    (f:% / g:%):% == apply(g,f)

changed:
-Prop(R:AbelianSemiGroup): Join(Monoidal R, CoercibleTo OutputForm) with
Prop(R:Ring): Join(Monoidal R, CoercibleTo OutputForm) with

changed:
-    (g:% * f:%):% ==
    apply(g:%, f:%):% ==

changed:
-    elt(f:%,g:%):% == per [dom(f)+dom(g),cod(f)+cod(g)]
    coerce(x:List None):% == per [1,1]
    (f:% * g:%):% == per [dom(f)+dom(g),cod(f)+cod(g)]

added:
    coerce: K -> %

added:
    apply: (%,%) -> %

added:
      dat(f)=0 => g
      dat(g)=0 => f

changed:
-    -- g*f : A^n -> A^p = g:A^m -> A^p * f:A^n -> A^m
    -- g f : A^n -> A^p = g:A^m -> A^p * f:A^n -> A^m

changed:
-    (g:% * f:%):% ==
-      cod(f) ~= dom(g) => error "arity"
    apply(g:%,f:%):% ==
      dom(g) ~= cod(f) => error "arity"

changed:
-      n:=dom(f)+1
-      m:=n+cod(f)
-      while m>n repeat
-        r := contract(r,n,m)
-        m:=m-1
      n:Integer:=dom(f)+1
      m:Integer:=dom(f)+cod(f)+1
      for i in 0..cod(f)-1 repeat
        r := contract(r,n,m-i)
        --(m1:=subtractIfCan(m,1)) case NNI =>m:=m1

changed:
-    1:% == per [1,1,kroneckerDelta()$T]
    coerce(x:List None):% == per [1,1,kroneckerDelta()$T]
    1:% == per [0,0,1]

    coerce(x:K):% == 1*x

added:
      print(p::OutputForm)

changed:
-    elt(f:%,g:%):% ==
    (f:% * g:%):% ==

changed:
-        [dom(f)+dom(g)+i for i in 1..cod(f)], _
-        [dom(f)+i for i in 1..dom(g)],        _
        [dom(f)+cod(f)+i for i in 1..dom(g)], _
        [dom(f)+i for i in 1..cod(f)],        _

added:
    (x:Integer * y:%):% == per [dom(y),cod(y),x*dat(y)]


removed:
-

changed:
-      #removeDuplicates([dom(y) for y in x])~=1 or
-        #removeDuplicates([cod(y) for y in x])~=1 => error "arity"
      #removeDuplicates([dom(y) for y in x]) ~= 1 or
        #removeDuplicates([cod(y) for y in x]) ~= 1 => error "arity"

changed:
-AB3:=(AB1/AB1)/AB1
AB3:=(AB1*AB1)*AB1

changed:
-test(AB3=AB1/(AB1/AB1))
test(AB3=AB1*(AB1*AB1))

changed:
-AB11:=A1 B1
AB11:=A1*B1

changed:
-BA11:= B1 A1
BA11:= B1*A1

changed:
-AB := A B
AB := A*B

changed:
-BA := B A
BA := B*A

changed:
-XB21:=X2 B1
XB21:=X2*B1

changed:
-XY22:=X2 Y2
XY22:=X2*Y2

changed:
-YX22:=Y2 X2
YX22:=Y2*X2

changed:
-\end{axiom}
test((A*A)*A=A*(A*A))
test((B*B)*B=B*(B*B))
test((A*B)*A=A*(B*A))

\end{axiom}

Linear transformations (operators) over n-dimensional cartesian vector spaces over a commutative ring . 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.

We try to start the right way by defining the concept of a monoidal category.

spad

)abbrev category MONAL Monoidal
)abbrev category MONAL Monoidal
Monoidal(R:AbelianSemiGroup):Category == Monoid with
    dom: % -> R
      ++ domain
    cod: % -> R
      ++ co-domain
    _/: (%,%) -> %
      ++ vertical composition f/g = g*f
    apply:(%,%) -> %
      ++ horizontal composition
    coerce:(x:List None) -> %
      ++ identity for composition
  add
    (f:% / g:%):% == apply(g,f)

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/6761162726479353213-25px001.spad using 
      old system compiler.
   MONAL abbreviates category Monoidal 
   MONAL abbreviates category Monoidal 
------------------------------------------------------------------------
   initializing NRLIB MONAL for Monoidal 
   compiling into NRLIB MONAL 

;;;     ***       |Monoidal| REDEFINED
Time: 0.03 SEC.

   MONAL- abbreviates domain Monoidal& 
------------------------------------------------------------------------
   initializing NRLIB MONAL- for Monoidal& 
   compiling into NRLIB MONAL- 
   compiling exported / : (S,S) -> S
Time: 0.33 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |Monoidal&| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor Monoidal&
      Time: 0.33 seconds

   finalizing NRLIB MONAL- 
   Processing Monoidal& for Browser database:
--------(dom (R %))---------
--------(cod (R %))---------
--------(/ (% % %))---------
--->-->Monoidal&((/ (% % %))): Improper first word in comments: vertical
"vertical composition \\spad{f/g} = \\spad{g*f}"
--------(apply (% % %))---------
--------(coerce (% (List (None))))---------
--->-->Monoidal&((coerce (% (List (None))))): Improper first word in comments: identity
"identity for composition"
--->-->Monoidal&(constructor): Not documented!!!!
--->-->Monoidal&(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/MONAL-.NRLIB/MONAL-.lsp" (written 09 APR 2011 10:14:08 PM):

; /var/zope2/var/LatexWiki/MONAL-.NRLIB/MONAL-.fasl written
; compilation finished in 0:00:00.066
------------------------------------------------------------------------
   Monoidal& is now explicitly exposed in frame initial 
   Monoidal& will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MONAL-.NRLIB/MONAL-
   finalizing NRLIB MONAL 
   Processing Monoidal for Browser database:
--------(dom (R %))---------
--------(cod (R %))---------
--------(/ (% % %))---------
--->-->Monoidal((/ (% % %))): Improper first word in comments: vertical
"vertical composition \\spad{f/g} = \\spad{g*f}"
--------(apply (% % %))---------
--------(coerce (% (List (None))))---------
--->-->Monoidal((coerce (% (List (None))))): Improper first word in comments: identity
"identity for composition"
--->-->Monoidal(constructor): Not documented!!!!
--->-->Monoidal(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL.lsp" (written 09 APR 2011 10:14:08 PM):

; /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL.fasl written
; compilation finished in 0:00:00.018
------------------------------------------------------------------------
   Monoidal is now explicitly exposed in frame initial 
   Monoidal will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL

   >> System error:
   The bounding indices 163 and 162 are bad for a sequence of length 162.
See also:
  The ANSI Standard, Glossary entry for "bounding index designator"
  The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

The initial object in this category is the domain Prop (Products and Permutations).

spad

)abbrev domain PROP Prop
Prop(R:Ring): Join(Monoidal R, CoercibleTo OutputForm) with
    _/:(R,R) -> %
  == add
    Rep == Record(domain:R,codomain:R)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    dom(f:%):R == rep(f).domain
    cod(f:%):R == rep(f).codomain

    coerce(f:%):OutputForm == dom(f)::OutputForm / cod(f)::OutputForm
    (f:R / g:R):% == per [f,g]
    apply(g:%, f:%):% ==
      cod(f) ~= dom(g) => error "arity"
      per [dom f,cod g]
    coerce(x:List None):% == per [1,1]
    (f:% * g:%):% == per [dom(f)+dom(g),cod(f)+cod(g)]
    -- preserves arity
    (f:% + g:%):% ==
      dom(f)~=dom(g) or cod(g) ~= cod(g) => error "arity"
      f

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/2209711262636362952-25px002.spad using 
      old system compiler.
   PROP abbreviates domain Prop 
------------------------------------------------------------------------
   initializing NRLIB PROP for Prop 
   compiling into NRLIB PROP 
   compiling local rep : $ -> Record(domain: R,codomain: R)
      PROP;rep is replaced by x 
Time: 0.04 SEC.

   compiling local per : Record(domain: R,codomain: R) -> $
      PROP;per is replaced by x 
Time: 0 SEC.

   compiling exported dom : $ -> R
Time: 0 SEC.

   compiling exported cod : $ -> R
Time: 0 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0.01 SEC.

   compiling exported / : (R,R) -> $
Time: 0 SEC.

   compiling exported apply : ($,$) -> $
Time: 0.01 SEC.

   compiling exported coerce : List None -> $
Time: 0.01 SEC.

   compiling exported * : ($,$) -> $
Time: 0 SEC.

   compiling local + : ($,$) -> $
Time: 0.01 SEC.

(time taken in buildFunctor:  10)

;;;     ***       |Prop| REDEFINED

;;;     ***       |Prop| REDEFINED
Time: 0.01 SEC.

   Cumulative Statistics for Constructor Prop
      Time: 0.09 seconds

   finalizing NRLIB PROP 
   Processing Prop for Browser database:
--->-->Prop((/ (% R R))): Not documented!!!!
--->-->Prop(constructor): Not documented!!!!
--->-->Prop(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/PROP.NRLIB/PROP.lsp" (written 09 APR 2011 10:14:09 PM):

; /var/zope2/var/LatexWiki/PROP.NRLIB/PROP.fasl written
; compilation finished in 0:00:00.247
------------------------------------------------------------------------
   Prop is now explicitly exposed in frame initial 
   Prop will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/PROP.NRLIB/PROP

   >> System error:
   The bounding indices 163 and 162 are bad for a sequence of length 162.
See also:
  The ANSI Standard, Glossary entry for "bounding index designator"
  The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

The LinearOperator? domain is Moniodal over NonNegativeInteger?

spad

)abbrev domain LIN LinearOperator
LinearOperator(dim:NonNegativeInteger,K:CommutativeRing): Exports == Implementation where
  NNI ==> NonNegativeInteger

  Exports ==> Join(Ring, BiModule(K,K), Monoidal NNI, CoercibleTo Prop NNI) with
    inp: List K -> %
      ++ incoming vector
    inp: List % -> %
    out: List K -> %
      ++ output vector
    out: List % -> %
    coerce: K -> %
    coerce: SquareMatrix(dim,K) -> %
    _*: (%,NonNegativeInteger) -> %
    arity: % -> Prop(NNI)
    apply: (%,%) -> %

  Implementation ==> add
    import List NNI
    T ==> CartesianTensor(1,dim,K)
    Rep == Record(domain:NNI, codomain:NNI, data:T)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    dom(f:%):NNI == rep(f).domain
    cod(f:%):NNI == rep(f).codomain

    dat(f:%):T == rep(f).data

    arity(f:%):Prop NNI == dom(f)/cod(f)
    coerce(f:%):Prop NNI == arity f

    0 == per [0,0,0]

    --
    -- f+g : A^{n+m} -> A^p = f:A^n -> A^p + g:A^m -> A^p
    --
    (f:% + g:%):% ==
      dat(f)=0 => g
      dat(g)=0 => f
      dom(f) ~= dom(g) or cod(f) ~= cod(g) => error "arity"
      per [dom(f),cod(f),dat(f)+dat(g)]

    (f:% - g:%):% ==
      dom(f) ~= dom(f) or cod(g) ~= cod(g) => error "arity"
      per [dom(f),cod(f),dat(f)-dat(g)]

    --
    -- f/g : A^n -> A^p = f:A^n -> A^m / g:A^m -> A^p
    -- g f : A^n -> A^p = g:A^m -> A^p * f:A^n -> A^m
    --
    apply(g:%,f:%):% ==
      dom(g) ~= cod(f) => error "arity"
      r:T := product(dat(f), dat(g))
      n:Integer:=dom(f)+1
      m:Integer:=dom(f)+cod(f)+1
      for i in 0..cod(f)-1 repeat
        r := contract(r,n,m-i)
        --(m1:=subtractIfCan(m,1)) case NNI =>m:=m1
      per [dom f,cod g,r]

    coerce(x:List None):% == per [1,1,kroneckerDelta()$T]
    1:% == per [0,0,1]

    coerce(x:K):% == 1*x

    -- repeated composition
    (f:% ^ p:NNI):% ==
      cod(f) ~= dom(f) => error "arity"
      q:=subtractIfCan(p,1)
      q case NNI => f^q * f
      1

    -- repeated sum
    (f:% * p:NNI):% ==
      print(p::OutputForm)
      q:=subtractIfCan(p,1)
      q case NNI => f*q + f
      0

    (f:% * g:%):% ==
      r:T := product(dat f,dat g)
      -- dom(f) + cod(f) + dom(g) + cod(g)
      p:List Integer := concat                _
        [[i for i in 1..dom(f)],              _
        [dom(f)+cod(f)+i for i in 1..dom(g)], _
        [dom(f)+i for i in 1..cod(f)],        _
        [dom(f)+dom(g)+cod(f)+i for i in 1..cod(g)]]
      -- dom(f) + dom(g) + cod(f) + cod(g)
      per [dom(f)+dom(g),cod(f)+cod(g),reindex(r,p)]

    (x:% = y:%):Boolean ==
      dom(x) ~= dom(y) or cod(x) ~= cod(y) => error "arity"
      dat(x) = dat(y)

    (x:K * y:%):% == per [dom(y),cod(y),x*dat(y)]

    (x:% * y:K):% == per [dom(x),cod(x),dat(x)*y]

    (x:Integer * y:%):% == per [dom(y),cod(y),x*dat(y)]

    inp(x:List K):% == per [1,0,entries(x)::T]
    inp(x:List %):% ==
      #removeDuplicates([dom(y) for y in x]) ~= 1 or
        #removeDuplicates([cod(y) for y in x]) ~= 1 => error "arity"
      per [dom(first x)+1,cod(first x),[dat(y) for y in x]::T]$Rep

    out(x:List K):% == per [0,1,entries(x)::T]

    out(x:List %):% ==
      #removeDuplicates([dom(y) for y in x])~=1 or
        #removeDuplicates([cod(y) for y in x])~=1 => error "arity"
      per [dom(first x),cod(first x)+1,[dat(y) for y in x]::T]$Rep

    coerce(x:%):OutputForm == (dat(x))::OutputForm

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/2457767226733576822-25px003.spad using 
      old system compiler.
   LIN abbreviates domain LinearOperator 
------------------------------------------------------------------------
   initializing NRLIB LIN for LinearOperator 
   compiling into NRLIB LIN 
   importing List NonNegativeInteger
   processing macro definition T$ ==> CartesianTensor(One,dim,K) 
   compiling local rep : $ -> Record(domain: NonNegativeInteger,codomain: NonNegativeInteger,data: CartesianTensor(One,dim,K))
      LIN;rep is replaced by x 
Time: 0.05 SEC.

   compiling local per : Record(domain: NonNegativeInteger,codomain: NonNegativeInteger,data: CartesianTensor(One,dim,K)) -> $
      LIN;per is replaced by x 
Time: 0 SEC.

   compiling exported dom : $ -> NonNegativeInteger
Time: 0 SEC.

   compiling exported cod : $ -> NonNegativeInteger
Time: 0 SEC.

   compiling local dat : $ -> CartesianTensor(One,dim,K)
Time: 0.01 SEC.

   compiling exported arity : $ -> Prop NonNegativeInteger
Time: 0 SEC.

   compiling exported coerce : $ -> Prop NonNegativeInteger
Time: 0 SEC.

   compiling exported Zero : () -> $
Time: 0.01 SEC.

   compiling exported + : ($,$) -> $
Time: 0.01 SEC.

   compiling exported - : ($,$) -> $
Time: 0 SEC.

   compiling exported apply : ($,$) -> $
Time: 0.01 SEC.

   compiling exported coerce : List None -> $
Time: 0.01 SEC.

   compiling exported One : () -> $
Time: 0 SEC.

   compiling exported coerce : K -> $
Time: 0 SEC.

   compiling exported ^ : ($,NonNegativeInteger) -> $
Time: 0.01 SEC.

   compiling exported * : ($,NonNegativeInteger) -> $
Time: 0.01 SEC.

   compiling exported * : ($,$) -> $
Time: 0.05 SEC.

   compiling exported = : ($,$) -> Boolean
Time: 0 SEC.

   compiling exported * : (K,$) -> $
Time: 0.01 SEC.

   compiling exported * : ($,K) -> $
Time: 0 SEC.

   compiling exported * : (Integer,$) -> $
Time: 0 SEC.

   compiling exported inp : List K -> $
Time: 0.05 SEC.

   compiling exported inp : List $ -> $
Time: 0.25 SEC.

   compiling exported out : List K -> $
Time: 0.01 SEC.

   compiling exported out : List $ -> $
Time: 0.02 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |LinearOperator| REDEFINED

;;;     ***       |LinearOperator| REDEFINED
Time: 0.02 SEC.

   Warnings: 
      [1] not known that (Ring) is of mode (CATEGORY domain (SIGNATURE quo ($ $ $)) (SIGNATURE rem ($ $ $)) (SIGNATURE gcd ($ $ $)) (SIGNATURE divide ((Record (: quotient $) (: remainder $)) $ $)) (SIGNATURE exquo ((Union $ failed) $ $)) (SIGNATURE shift ($ $ (Integer))) (SIGNATURE random ($ $)))
      [2] arity: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE quo ($ $ $)) (SIGNATURE rem ($ $ $)) (SIGNATURE gcd ($ $ $)) (SIGNATURE divide ((Record (: quotient $) (: remainder $)) $ $)) (SIGNATURE exquo ((Union $ failed) $ $)) (SIGNATURE shift ($ $ (Integer))) (SIGNATURE random ($ $)))

   Cumulative Statistics for Constructor LinearOperator
      Time: 0.53 seconds

   finalizing NRLIB LIN 
   Processing LinearOperator for Browser database:
--------(inp (% (List K)))---------
--->-->LinearOperator((inp (% (List %)))): Not documented!!!!
--------(out (% (List K)))---------
--->-->LinearOperator((out (% (List %)))): Not documented!!!!
--->-->LinearOperator((coerce (% K))): Not documented!!!!
--->-->LinearOperator((coerce (% (SquareMatrix dim K)))): Not documented!!!!
--->-->LinearOperator((* (% % (NonNegativeInteger)))): Not documented!!!!
--->-->LinearOperator((arity ((Prop NNI) %))): Not documented!!!!
--->-->LinearOperator((apply (% % %))): Not documented!!!!
--->-->LinearOperator(constructor): Not documented!!!!
--->-->LinearOperator(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/LIN.NRLIB/LIN.lsp" (written 09 APR 2011 10:14:10 PM):

; /var/zope2/var/LatexWiki/LIN.NRLIB/LIN.fasl written
; compilation finished in 0:00:00.439
------------------------------------------------------------------------
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/LIN.NRLIB/LIN

   >> System error:
   The bounding indices 163 and 162 are bad for a sequence of length 162.
See also:
  The ANSI Standard, Glossary entry for "bounding index designator"
  The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

Construction operators: input and output

axiom

L:=LIN(2,FRAC POLY INT)

$\label{eq1}\hbox{\axiomType{LinearOperator}\ } (2, \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })))$

(1)

Type: Type

axiom

A1:L:=inp[script(a,[[1,i]]) for i in 1..2]

$\label{eq2}\left[{a_{1, \: 1}}, \:{a_{1, \: 2}}\right]$

(2)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity A1

$\label{eq3}1 \over 0$

(3)

Type: Prop(NonNegativeInteger?)

axiom

A2:L:=inp[script(a,[[2,i]]) for i in 1..2]

$\label{eq4}\left[{a_{2, \: 1}}, \:{a_{2, \: 2}}\right]$

(4)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

A:L:=inp[A1,A2]

$\label{eq5}\left[ \begin{array}{cc} {a_{1, \: 1}}&{a_{1, \: 2}} \ {a_{2, \: 1}}&{a_{2, \: 2}}$

(5)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity A

$\label{eq6}2 \over 0$

(6)

Type: Prop(NonNegativeInteger?)

axiom

B1:L:=out[script(b,[[],[1,i]]) for i in 1..2]

$\label{eq7}\left[{b^{1, \: 1}}, \:{b^{1, \: 2}}\right]$

(7)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity B1

$\label{eq8}0 \over 1$

(8)

Type: Prop(NonNegativeInteger?)

axiom

B2:L:=out[script(b,[[],[2,i]]) for i in 1..2]

$\label{eq9}\left[{b^{2, \: 1}}, \:{b^{2, \: 2}}\right]$

(9)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

B:L:=out[B1,B2]

$\label{eq10}\left[ \begin{array}{cc} {b^{1, \: 1}}&{b^{1, \: 2}} \ {b^{2, \: 1}}&{b^{2, \: 2}}$

(10)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity B

$\label{eq11}0 \over 2$

(11)

Type: Prop(NonNegativeInteger?)

Composition

$g/f : A^n \to A^{m+p} = f:A^n \to A^m <em> g:A^n \to A^p$

axiom

AB2 := A2 / B2; AB2::OutputForm = A2::OutputForm / B2::OutputForm

$\label{eq12}\begin{array}{@{}l} \displaystyle {\left[ \begin{array}{cc} {{b^{2, \: 1}}\ {a_{2, \: 1}}}&{{b^{2, \: 2}}\ {a_{2, \: 1}}} \ {{b^{2, \: 1}}\ {a_{2, \: 2}}}&{{b^{2, \: 2}}\ {a_{2, \: 2}}}$

(12)

Type: Equation(OutputForm?)

axiom

arity(AB2)::OutputForm = arity(A2)::OutputForm / arity(B2)::OutputForm

$\label{eq13}{1 \over 1}={{1 \over 0}\over{0 \over 1}}$

(13)

Type: Equation(OutputForm?)

axiom

BA1 := B1 / A1; BA1::OutputForm = B1::OutputForm / A1::OutputForm

$\label{eq14}{{{b^{1, \: 2}}\ {a_{1, \: 2}}}+{{b^{1, \: 1}}\ {a_{1, \: 1}}}}={{\left[{b^{1, \: 1}}, \:{b^{1, \: 2}}\right]}\over{\left[{a_{1, \: 1}}, \:{a_{1, \: 2}}\right]}}$

(14)

Type: Equation(OutputForm?)

axiom

arity(BA1)::OutputForm = arity(B1)::OutputForm / arity(A1)::OutputForm

$\label{eq15}{0 \over 0}={{0 \over 1}\over{1 \over 0}}$

(15)

Type: Equation(OutputForm?)

axiom

AB1 := A1 / B1; AB1::OutputForm = A1::OutputForm / B1::OutputForm

$\label{eq16}\begin{array}{@{}l} \displaystyle {\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}} \ {{b^{1, \: 1}}\ {a_{1, \: 2}}}&{{b^{1, \: 2}}\ {a_{1, \: 2}}}$

(16)

Type: Equation(OutputForm?)

axiom

arity(AB1)::OutputForm = arity(A1)::OutputForm / arity(B1)::OutputForm

$\label{eq17}{1 \over 1}={{1 \over 0}\over{0 \over 1}}$

(17)

Type: Equation(OutputForm?)

Powers

axiom

AB3:=(AB1*AB1)*AB1

$\label{eq18}\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{{b^{1, \: 1}}^3}\ {{a_{1, \: 1}}^3}}&{{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}\ {{a_{1, \: 1}}^3}} \ {{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}\ {{a_{1, \: 1}}^3}}&{{b^{1, \: 1}}\ {{b^{1, \: 2}}^2}\ {{a_{1, \: 1}}^3}}$

(18)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(AB3)

$\label{eq19}3 \over 3$

(19)

Type: Prop(NonNegativeInteger?)

axiom

test(AB3=AB1*(AB1*AB1))

$\label{eq20} \mbox{\rm true}$

(20)

Type: Boolean

axiom

test(AB3=AB1^3)

$\label{eq21} \mbox{\rm true}$

(21)

Type: Boolean

Sums

$f+g : A^n \to A^m = f:A^n \to A^m + g:A^n \to A^m$

axiom

A12s := A1 + A2; A12s::OutputForm = A1::OutputForm + A2::OutputForm

$\label{eq22}{\left[{{a_{2, \: 1}}+{a_{1, \: 1}}}, \:{{a_{2, \: 2}}+{a_{1, \: 2}}}\right]}={{\left[{a_{1, \: 1}}, \:{a_{1, \: 2}}\right]}+{\left[{a_{2, \: 1}}, \:{a_{2, \: 2}}\right]}}$

(22)

Type: Equation(OutputForm?)

axiom

arity(A12s)::OutputForm = arity(A1)::OutputForm + arity(A2)::OutputForm

$\label{eq23}{1 \over 0}={{1 \over 0}+{1 \over 0}}$

(23)

Type: Equation(OutputForm?)

axiom

B12s := B1 + B2; B12s::OutputForm = B1::OutputForm + B2::OutputForm

$\label{eq24}{\left[{{b^{2, \: 1}}+{b^{1, \: 1}}}, \:{{b^{2, \: 2}}+{b^{1, \: 2}}}\right]}={{\left[{b^{1, \: 1}}, \:{b^{1, \: 2}}\right]}+{\left[{b^{2, \: 1}}, \:{b^{2, \: 2}}\right]}}$

(24)

Type: Equation(OutputForm?)

axiom

arity(B12s)::OutputForm = arity(B1)::OutputForm + arity(B2)::OutputForm

$\label{eq25}{0 \over 1}={{0 \over 1}+{0 \over 1}}$

(25)

Type: Equation(OutputForm?)

Multiplication

axiom

A3s:=(A1+A1)+A1

$\label{eq26}\left[{3 \ {a_{1, \: 1}}}, \:{3 \ {a_{1, \: 2}}}\right]$

(26)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(A3s)

$\label{eq27}1 \over 0$

(27)

Type: Prop(NonNegativeInteger?)

axiom

test(A3s=A1+(A1+A1))

$\label{eq28} \mbox{\rm true}$

(28)

Type: Boolean

axiom

test(A3s=A1*3)

   3
   2
   1
   0

$\label{eq29} \mbox{\rm true}$

(29)

Type: Boolean

axiom

B3s:=(B1+B1)+B1

$\label{eq30}\left[{3 \ {b^{1, \: 1}}}, \:{3 \ {b^{1, \: 2}}}\right]$

(30)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(B3s)

$\label{eq31}0 \over 1$

(31)

Type: Prop(NonNegativeInteger?)

axiom

test(B3s=B1+(B1+B1))

$\label{eq32} \mbox{\rm true}$

(32)

Type: Boolean

axiom

test(B3s=B1*3)

   3
   2
   1
   0

$\label{eq33} \mbox{\rm true}$

(33)

Type: Boolean

Expected error

axiom

B1A1:=B1*A1

$\label{eq34}\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}} \ {{b^{1, \: 1}}\ {a_{1, \: 2}}}&{{b^{1, \: 2}}\ {a_{1, \: 2}}}$

(34)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(B1A1)

$\label{eq35}1 \over 1$

(35)

Type: Prop(NonNegativeInteger?)

Product

axiom

AB11:=A1*B1

$\label{eq36}\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}} \ {{b^{1, \: 1}}\ {a_{1, \: 2}}}&{{b^{1, \: 2}}\ {a_{1, \: 2}}}$

(36)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(AB11)

$\label{eq37}1 \over 1$

(37)

Type: Prop(NonNegativeInteger?)

axiom

BA11:= B1*A1

$\label{eq38}\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}} \ {{b^{1, \: 1}}\ {a_{1, \: 2}}}&{{b^{1, \: 2}}\ {a_{1, \: 2}}}$

(38)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(BA11)

$\label{eq39}1 \over 1$

(39)

Type: Prop(NonNegativeInteger?)

axiom

AB := A*B

$\label{eq40}\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}} \ {{b^{2, \: 1}}\ {a_{1, \: 1}}}&{{b^{2, \: 2}}\ {a_{1, \: 1}}}$

(40)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(AB)

$\label{eq41}2 \over 2$

(41)

Type: Prop(NonNegativeInteger?)

axiom

BA := B*A

$\label{eq42}\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}} \ {{b^{2, \: 1}}\ {a_{1, \: 1}}}&{{b^{2, \: 2}}\ {a_{1, \: 1}}}$

(42)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(BA)

$\label{eq43}2 \over 2$

(43)

Type: Prop(NonNegativeInteger?)

axiom

X2:L:=inp[inp([script(x,[[i,j]]) for j in 1..2])$L for i in 1..2]

$\label{eq44}\left[ \begin{array}{cc} {x_{1, \: 1}}&{x_{1, \: 2}} \ {x_{2, \: 1}}&{x_{2, \: 2}}$

(44)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

XB21:=X2*B1

$\label{eq45}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {x_{1, \: 1}}}&{{b^{1, \: 2}}\ {x_{1, \: 1}}} \ {{b^{1, \: 1}}\ {x_{1, \: 2}}}&{{b^{1, \: 2}}\ {x_{1, \: 2}}}$

(45)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(XB21)

$\label{eq46}2 \over 1$

(46)

Type: Prop(NonNegativeInteger?)

axiom

Y2:L:=out[out([script(y,[[],[i,j]]) for j in 1..2])$L for i in 1..2]

$\label{eq47}\left[ \begin{array}{cc} {y^{1, \: 1}}&{y^{1, \: 2}} \ {y^{2, \: 1}}&{y^{2, \: 2}}$

(47)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

XY22:=X2*Y2

$\label{eq48}\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{y^{1, \: 1}}\ {x_{1, \: 1}}}&{{y^{1, \: 2}}\ {x_{1, \: 1}}} \ {{y^{2, \: 1}}\ {x_{1, \: 1}}}&{{y^{2, \: 2}}\ {x_{1, \: 1}}}$

(48)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(XY22)

$\label{eq49}2 \over 2$

(49)

Type: Prop(NonNegativeInteger?)

axiom

YX22:=Y2*X2

$\label{eq50}\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{y^{1, \: 1}}\ {x_{1, \: 1}}}&{{y^{1, \: 2}}\ {x_{1, \: 1}}} \ {{y^{2, \: 1}}\ {x_{1, \: 1}}}&{{y^{2, \: 2}}\ {x_{1, \: 1}}}$

(50)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(XY22)

$\label{eq51}2 \over 2$

(51)

Type: Prop(NonNegativeInteger?)

axiom

test((A*A)*A=A*(A*A))

$\label{eq52} \mbox{\rm true}$

(52)

Type: Boolean

axiom

test((B*B)*B=B*(B*B))

$\label{eq53} \mbox{\rm true}$

(53)

Type: Boolean

axiom

test((A*B)*A=A*(B*A))

$\label{eq54} \mbox{\rm true}$

(54)

Type: Boolean

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]

$\label{eq55}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {w_{1, \: 1}^{1}}&{w_{1, \: 2}^{1}} \ {w_{2, \: 1}^{1}}&{w_{2, \: 2}^{1}}$

(55)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

WB1:=W B1

   >> Error detected within library code:
   arity

product and composition --Bill Page, Wed, 09 Mar 2011 18:53:36 -0800 reply

Another kind of diagram:

  ___ _       _        _  _
  ___ _)  =   _\_ _    _\/_
  _/          ___ _)   _/\_

Linear operators as morphisms --Bill Page, Thu, 10 Mar 2011 09:42:33 -0800 reply

If linear operators really are to be morphisms (in the sense of category theory) then they must have a domain and a co-domain that are vector spaces, not just an in-degree and out-degree. E.g.:

  Rep == Record(Dom:VectorSpace, Cod:VectorSpace, t:T)

But VectorSpace? is a category which would make Dom and Cod domains. The domains that currently satisfy VectorSpace? in Axiom are rather limited and seem oddly focused on number theory (finite fields). It is a good thing however that DirectProduct? is a conditional member of this cateogry.

axiom

DirectProduct(2,FRAC INT) has VectorSpace(FRAC INT)

$\label{eq56} \mbox{\rm true}$

(56)

Type: Boolean

Unfortunately:

axiom

DirectProduct(2,DirectProduct(2,FRAC INT)) has VectorSpace(FRAC INT)

$\label{eq57} \mbox{\rm false}$

(57)

Type: Boolean

The problem with VectorSpace? is that the domain [Vector]? is not a member of this category instead it satisfies VectoryCategory?.

Is it possible to treat tensors from CartesianTensor? as maps from VectorSpace? to VectorSpace?? We would like for example:

  T:DirectProduct(dim,DirectProduct(dim,FRAC INT)) -> DirectProduct(dim,FRAC INT))

To be a linear operator with two inputs and one output.

Modules --Bill Page, Thu, 10 Mar 2011 16:03:13 -0800 reply

A [FreeModule]? over a [Field]? is a VectorSpace? unfortunately this is not currently understood by Axiom:

axiom

FreeModule(Fraction Integer,OrderedVariableList [e1,e1]) has VectorSpace(Fraction Integer)

$\label{eq58} \mbox{\rm true}$

(58)

Type: Boolean