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LinearOperator

last edited 2 years ago by Bill Page

Edit detail for LinearOperator revision 9 of 63

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Editor: Bill Page Time: 2011/03/10 09:42:42 GMT-8
Note: Linear operators as morphisms

added:

From BillPage Thu Mar 10 09:42:33 -0800 2011
From: Bill Page
Date: Thu, 10 Mar 2011 09:42:33 -0800
Subject: Linear operators as morphisms
Message-ID: <20110310094233-0800@axiom-wiki.newsynthesis.org>

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.
\begin{axiom}
DirectProduct(2,FRAC INT) has VectorSpace(FRAC INT)
\end{axiom}
Unfortunately:
\begin{axiom}
DirectProduct(2,DirectProduct(2,FRAC INT)) has VectorSpace(FRAC INT)
\end{axiom}

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.

Linear transformations (operators) over n-dimensional cartesian vector spaces oveer 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.

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) -> %
    _*: (%,NonNegativeInteger) -> %
  == 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]

    --
    -- f*g : A^n -> A^{m+p} = f:A^n -> A^m * g:A^n -> A^p
    --
    (x:% * y:%):% ==
      output("* rep(x).n",rep(x).n::OutputForm)$OutputPackage
      output("* rep(y).n",rep(y).n::OutputForm)$OutputPackage
      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]

    -- repeated product
    ((x:%)^(p:NonNegativeInteger)):% ==
       q:=subtractIfCan(p,1)
       q case NonNegativeInteger => x^q * x
       per [rep(x).n,0,1]

    --
    -- f+g : A^{n+m} -> A^p = f:A^n -> A^p + g:A^m -> A^p
    --
    (x:% + y:%):% ==
      output("+ rep(x).m",rep(x).m::OutputForm)$OutputPackage
      output("+ rep(y).m",rep(y).m::OutputForm)$OutputPackage
      rep(x).m ~= rep(y).m  => error "arity"
      output("+ rep(x).n",rep(x).n::OutputForm)$OutputPackage
      output("+ rep(y).n",rep(y).n::OutputForm)$OutputPackage
      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)$OutputPackage
        output("+ rep(y).m",rep(y).m::OutputForm)$OutputPackage
        r := contract(contract(du,1,r,rep(y).m+1),1,rep(y).m+2)
        --output("rank",rank(r)::OutputForm)$OutputPackage
      per [rep(x).n+rep(y).n,rep(y).m,r]

    -- repeated sum
    (x:% * p:NonNegativeInteger):% ==
      q:=subtractIfCan(p,1)
      q case NonNegativeInteger => x*q + x
      per [0,rep(x).m,1]  -- need identity for + !!!

    (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/6154772542123042263-25px001.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,m: NonNegativeInteger,t: CartesianTensor(One,dim,K))
      LIN;rep is replaced by x 
Time: 0.09 SEC.

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

   compiling exported arity : $ -> DirectProduct(2,NonNegativeInteger)
Time: 0.20 SEC.

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

   compiling exported + : ($,$) -> $
Time: 0.02 SEC.

   compiling exported - : ($,$) -> $
Time: 0.02 SEC.

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

   compiling exported * : ($,$) -> $
Time: 0.03 SEC.

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

   compiling exported + : ($,$) -> $
Time: 0.03 SEC.

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

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

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

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

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

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

   compiling exported inp : List K -> $
Time: 0.08 SEC.

   compiling exported inp : List $ -> $
Time: 0.12 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:  10)

;;;     ***       |LinearOperator| REDEFINED

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

   Cumulative Statistics for Constructor LinearOperator
      Time: 0.69 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((* (% % (NonNegativeInteger)))): Not documented!!!!
--->-->LinearOperator(constructor): Not documented!!!!
--->-->LinearOperator(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/LIN.NRLIB/LIN.lsp" (written 09 MAR 2011 07:00:54 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$;10| ...)
; compiling (DEFUN |LIN;*;$Nni$;11| ...)
; compiling (DEFUN |LIN;=;2$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 0:00:00.884
------------------------------------------------------------------------
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/LIN.NRLIB/LIN

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: Domain

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

$\label{eq3}\left[ 1, \: 0 \right]$

(3)

Type: DirectProduct?(2,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

$\label{eq6}\left[ 2, \: 0 \right]$

(6)

Type: DirectProduct?(2,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

$\label{eq8}\left[ 0, \: 1 \right]$

(8)

Type: DirectProduct?(2,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

$\label{eq11}\left[ 0, \: 2 \right]$

(11)

Type: DirectProduct?(2,NonNegativeInteger?)

Products

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

axiom

A12p := A1 * A2; A12p::OutputForm = A1::OutputForm * A2::OutputForm

   * rep(x).n 1
   * rep(y).n 1

$\label{eq12}\begin{array}{@{}l} \displaystyle {\left[{{2 \ {a_{1, \: 1}}\ {a_{2, \: 2}}}+{2 \ {a_{1, \: 1}}\ {a_{2, \: 1}}}}, \:{{2 \ {a_{1, \: 2}}\ {a_{2, \: 2}}}+{2 \ {a_{1, \: 2}}\ {a_{2, \: 1}}}}\right]}= \ \ \displaystyle {{\left[{a_{1, \: 1}}, \:{a_{1, \: 2}}\right]}\ {\left[{a_{2, \: 1}}, \:{a_{2, \: 2}}\right]}}$

(12)

Type: Equation(OutputForm?)

axiom

arity(A12p)::OutputForm = arity(A1)::OutputForm * arity(A2)::OutputForm

$\label{eq13}{\left[ 1, \: 0 \right]}={{\left[ 1, \: 0 \right]}\ {\left[ 1, \: 0 \right]}}$

(13)

Type: Equation(OutputForm?)

axiom

B12p := B1 * B2; B12p::OutputForm = B1::OutputForm * B2::OutputForm

   * rep(x).n 0
   * rep(y).n 0

$\label{eq14}\begin{array}{@{}l} \displaystyle {\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {b^{2, \: 1}}}&{{b^{1, \: 1}}\ {b^{2, \: 2}}} \ {{b^{1, \: 2}}\ {b^{2, \: 1}}}&{{b^{1, \: 2}}\ {b^{2, \: 2}}}$

(14)

Type: Equation(OutputForm?)

axiom

arity(B12p)::OutputForm = arity(B1)::OutputForm * arity(B2)::OutputForm

$\label{eq15}{\left[ 0, \: 2 \right]}={{\left[ 0, \: 1 \right]}\ {\left[ 0, \: 1 \right]}}$

(15)

Type: Equation(OutputForm?)

Powers

axiom

A3p:=(A1*A1)*A1

   * rep(x).n 1
   * rep(y).n 1
   * rep(x).n 1
   * rep(y).n 1

$\label{eq16}\begin{array}{@{}l} \displaystyle \left[{{4 \ {a_{1, \: 1}}\ {{a_{1, \: 2}}^2}}+{8 \ {{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}+{4 \ {{a_{1, \: 1}}^3}}}, \: \right. \ \ \displaystyle \left.{{4 \ {{a_{1, \: 2}}^3}}+{8 \ {a_{1, \: 1}}\ {{a_{1, \: 2}}^2}}+{4 \ {{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}}\right]$

(16)

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

axiom

arity(A3p)

$\label{eq17}\left[ 1, \: 0 \right]$

(17)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

test(A3p=A1*(A1*A1))

   * rep(x).n 1
   * rep(y).n 1
   * rep(x).n 1
   * rep(y).n 1

$\label{eq18} \mbox{\rm true}$

(18)

Type: Boolean

axiom

test(A3p=A1^3)

   * rep(x).n 1
   * rep(y).n 1
   * rep(x).n 1
   * rep(y).n 1

$\label{eq19} \mbox{\rm true}$

(19)

Type: Boolean

axiom

B3p:=(B1*B1)*B1

   * rep(x).n 0
   * rep(y).n 0
   * rep(x).n 0
   * rep(y).n 0

$\label{eq20}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {{b^{1, \: 1}}^3}&{{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}} \ {{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}&{{b^{1, \: 1}}\ {{b^{1, \: 2}}^2}}$

(20)

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

axiom

arity(B3p)

$\label{eq21}\left[ 0, \: 3 \right]$

(21)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

test(B3p=B1*(B1*B1))

   * rep(x).n 0
   * rep(y).n 0
   * rep(x).n 0
   * rep(y).n 0

$\label{eq22} \mbox{\rm true}$

(22)

Type: Boolean

axiom

test(B3p=B1^3)

   * rep(x).n 0
   * rep(y).n 0
   * rep(x).n 0
   * rep(y).n 0

$\label{eq23} \mbox{\rm true}$

(23)

Type: Boolean

Sums

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

axiom

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

   + rep(x).m 0
   + rep(y).m 0
   + rep(x).n 1
   + rep(y).n 1

$\label{eq24}\begin{array}{@{}l} \displaystyle {\left[ \begin{array}{cc} {{a_{1, \: 1}}\ {a_{2, \: 1}}}&{{a_{1, \: 1}}\ {a_{2, \: 2}}} \ {{a_{1, \: 2}}\ {a_{2, \: 1}}}&{{a_{1, \: 2}}\ {a_{2, \: 2}}}$

(24)

Type: Equation(OutputForm?)

axiom

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

$\label{eq25}{\left[ 2, \: 0 \right]}={{\left[ 1, \: 0 \right]}+{\left[ 1, \: 0 \right]}}$

(25)

Type: Equation(OutputForm?)

axiom

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

   + rep(x).m 1
   + rep(y).m 1
   + rep(x).n 0
   + rep(y).n 0
   + rank 2
   + rep(y).m 1

$\label{eq26}\begin{array}{@{}l} \displaystyle {\left[{{{b^{1, \: 2}}\ {b^{2, \: 2}}}+{{b^{1, \: 1}}\ {b^{2, \: 1}}}}, \:{{{b^{1, \: 2}}\ {b^{2, \: 2}}}+{{b^{1, \: 1}}\ {b^{2, \: 1}}}}\right]}= \ \ \displaystyle {{\left[{b^{1, \: 1}}, \:{b^{1, \: 2}}\right]}+{\left[{b^{2, \: 1}}, \:{b^{2, \: 2}}\right]}}$

(26)

Type: Equation(OutputForm?)

axiom

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

$\label{eq27}{\left[ 0, \: 1 \right]}={{\left[ 0, \: 1 \right]}+{\left[ 0, \: 1 \right]}}$

(27)

Type: Equation(OutputForm?)

Multiplication

axiom

A3s:=(A1+A1)+A1

   + rep(x).m 0
   + rep(y).m 0
   + rep(x).n 1
   + rep(y).n 1
   + rep(x).m 0
   + rep(y).m 0
   + rep(x).n 2
   + rep(y).n 1

$\label{eq28}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {{a_{1, \: 1}}^3}&{{{a_{1, \: 1}}^2}\ {a_{1, \: 2}}} \ {{{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}&{{a_{1, \: 1}}\ {{a_{1, \: 2}}^2}}$

(28)

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

axiom

arity(A3s)

$\label{eq29}\left[ 3, \: 0 \right]$

(29)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

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

   + rep(x).m 0
   + rep(y).m 0
   + rep(x).n 1
   + rep(y).n 1
   + rep(x).m 0
   + rep(y).m 0
   + rep(x).n 1
   + rep(y).n 2

$\label{eq30} \mbox{\rm true}$

(30)

Type: Boolean

axiom

test(A3s=A1*3)

   + rep(x).m 0
   + rep(y).m 0
   + rep(x).n 0
   + rep(y).n 1
   + rep(x).m 0
   + rep(y).m 0
   + rep(x).n 1
   + rep(y).n 1
   + rep(x).m 0
   + rep(y).m 0
   + rep(x).n 2
   + rep(y).n 1

$\label{eq31} \mbox{\rm true}$

(31)

Type: Boolean

axiom

B3s:=(B1+B1)+B1

   + rep(x).m 1
   + rep(y).m 1
   + rep(x).n 0
   + rep(y).n 0
   + rank 2
   + rep(y).m 1
   + rep(x).m 1
   + rep(y).m 1
   + rep(x).n 0
   + rep(y).n 0
   + rank 2
   + rep(y).m 1

$\label{eq32}\begin{array}{@{}l} \displaystyle \left[{{{b^{1, \: 2}}^3}+{{b^{1, \: 1}}\ {{b^{1, \: 2}}^2}}+{{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}+{{b^{1, \: 1}}^3}}, \: \right. \ \ \displaystyle \left.{{{b^{1, \: 2}}^3}+{{b^{1, \: 1}}\ {{b^{1, \: 2}}^2}}+{{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}+{{b^{1, \: 1}}^3}}\right]$

(32)

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

axiom

arity(B3s)

$\label{eq33}\left[ 0, \: 1 \right]$

(33)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

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

   + rep(x).m 1
   + rep(y).m 1
   + rep(x).n 0
   + rep(y).n 0
   + rank 2
   + rep(y).m 1
   + rep(x).m 1
   + rep(y).m 1
   + rep(x).n 0
   + rep(y).n 0
   + rank 2
   + rep(y).m 1

$\label{eq34} \mbox{\rm true}$

(34)

Type: Boolean

axiom

test(B3s=B1*3)

   + rep(x).m 1
   + rep(y).m 1
   + rep(x).n 0
   + rep(y).n 0
   + rank 1
   + rep(y).m 1

   >> Error detected within library code:
   Improper index for contraction

Expected error

axiom

B1A1:=B1*A1

   * rep(x).n 0
   * rep(y).n 1

   >> Error detected within library code:
   arity

Composition

axiom

AB11:=A1 B1

$\label{eq35}\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}} \ {{b^{1, \: 1}}\ {a_{1, \: 2}}}&{{b^{1, \: 2}}\ {a_{1, \: 2}}}$

(35)

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

axiom

arity(AB11)

$\label{eq36}\left[ 1, \: 1 \right]$

(36)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

BA11:= B1 A1

   yn 1
   xm 1
   rank 2

$\label{eq37}{{b^{1, \: 2}}\ {a_{1, \: 2}}}+{{b^{1, \: 1}}\ {a_{1, \: 1}}}$

(37)

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

axiom

arity(BA11)

$\label{eq38}\left[ 0, \: 0 \right]$

(38)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

AB := A B

$\label{eq39}\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}} \ {{b^{2, \: 1}}\ {a_{1, \: 1}}}&{{b^{2, \: 2}}\ {a_{1, \: 1}}}$

(39)

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

axiom

arity(AB)

$\label{eq40}\left[ 2, \: 2 \right]$

(40)

Type: DirectProduct?(2,NonNegativeInteger?)

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]

$\label{eq41}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {w_{1, \: 1}^{1}}&{w_{1, \: 2}^{1}} \ {w_{2, \: 1}^{1}}&{w_{2, \: 2}^{1}}$

(41)

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

axiom

WB1:=W B1

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

(42)

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

axiom

arity(WB1)

$\label{eq43}\left[ 3, \: 1 \right]$

(43)

Type: DirectProduct?(2,NonNegativeInteger?)

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)

$\label{eq1} \mbox{\rm true}$

(1)

Type: Boolean

Unfortunately:

axiom

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

$\label{eq2} \mbox{\rm false}$

(2)

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.