Edit detail for LinearOperator revision 32 of 63

Introduction

Bi-graded linear operators (transformations) over n-dimensional vector spaces on a commutative ring . Members of this domain are morphisms . Products, permutations and composition (grafting) of morphisms are 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.

See examples and documentation below

I would like you to make brief comments in the form at the bottom of this web page. For more detailed but related comments click discussion on the top menu.

Regards, Bill Page.

Source Code

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

Ref: http://en.wikipedia.org/wiki/PROP_(category_theory)

(1) -> <spad>

)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
--  add
--    (f:% / g:%):% == apply(g,f)</spad>

Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2643402771659714163-25px001.spad
      using system compiler.
   MONAL abbreviates category Monoidal 
------------------------------------------------------------------------
   initializing NRLIB MONAL for Monoidal 
   compiling into NRLIB MONAL 
Time: 0 SEC.

   finalizing NRLIB MONAL 
   Processing Monoidal for Browser database:
--->-->Monoidal(constructor): Not documented!!!!
--------(dom (R %))---------
--------(cod (R %))---------
--------(/ (% % %))---------
--->-->Monoidal((/ (% % %))): Improper first word in comments: vertical
"vertical composition \\spad{f/g} = \\spad{g*f}"
--------(apply (% % %))---------
--->-->Monoidal: Missing Description
; compiling file "/var/aw/var/LatexWiki/MONAL.NRLIB/MONAL.lsp" (written 01 NOV 2026 11:11:02 AM):

; wrote /var/aw/var/LatexWiki/MONAL.NRLIB/MONAL.fasl
; compilation finished in 0:00:00.012
------------------------------------------------------------------------
   Monoidal is now explicitly exposed in frame initial 
   Monoidal will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/MONAL.NRLIB/MONAL

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

)abbrev domain PROP Prop
Prop(): Exports == Implementation where
  R ==> NonNegativeInteger

  Exports ==> Join(Monoidal R, CoercibleTo OutputForm) with
    _/:(R,R) -> %

  Implementation ==> 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(gg:%,ff:%):% ==
      -- partial application from the right
      n:=subtractIfCan(dom ff,cod gg)
      f:=ff; n case R => f:=1^n*ff
      m:=subtractIfCan(cod gg, dom ff)
      g:=gg; m case R => g:=1^m*gg
      f/g
    (ff:% / gg:%):% ==
      -- partial application from the left
      n:=subtractIfCan(dom ff,cod gg)
      f:=ff; n case R => f:=ff*1^n
      m:=subtractIfCan(cod gg, dom ff)
      g:=gg; m case R => g:=gg*1^m
      --dom(g) ~= cod(f) => error "arity"
      per [dom f,cod g]
    1:% == per [0,0]
    (f:% * g:%):% == per [dom(f)+dom(g),cod(f)+cod(g)]
    0:% == per [0,0]
    -- preserves arity
    (f:% + g:%):% ==
      dom(f)~=dom(g) or cod(g) ~= cod(g) => error "arity"
      f

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3365105071098393884-25px002.spad
      using system compiler.
   PROP abbreviates domain Prop 
------------------------------------------------------------------------
   initializing NRLIB PROP for Prop 
   compiling into NRLIB PROP 
************* USER ERROR **********
available signatures for Rep: 
    NONE
NEED Rep: () -> ?
****** comp fails at level 1 with expression: ******
((DEF (|Rep|) (NIL)
  (|Record| (|:| |domain| (|NonNegativeInteger|))
            (|:| |codomain| (|NonNegativeInteger|)))))
****** level 1  ******
x:= (DEF (Rep) (NIL) (Record (: domain (NonNegativeInteger)) (: codomain (NonNegativeInteger))))
m:= $EmptyMode
f:=
((((~= #) (= #) (|coerce| #) (|latex| #) ...)))

   >> Apparent user error:
   unspecified error

The LinearOperator? domain is Moniodal over NonNegativeInteger?. It is the free algebra over Prop with n-generators.

Ref: http://en.wikipedia.org/wiki/Category_of_vector_spaces

• all members of this domain have the same dimension
• it might be useful (but much more complicated) to allow source and target dimensions to be different

)abbrev domain LIN LinearOperator
LinearOperator(dim:NNI,gen:OrderedFinite,K:CommutativeRing): Exports == Implementation where
  NNI ==> NonNegativeInteger
  T ==> CartesianTensor(1,dim,K)

  Exports ==> Join(Ring, BiModule(K,K), Monoidal NNI, RetractableTo K) with
    inp: List K -> %
      ++ incoming vector
    inp: List % -> %
    out: List K -> %
      ++ output vector
    out: List % -> %
    arity: % -> Prop
    basisVectors: () -> List %
    basisForms: () -> List %
    tensor: % -> T
    map: (K->K,%) -> %
    ravel: % -> List K
    unravel: (Prop,List K) -> %
    coerce:(x:List PositiveInteger) -> %
      ++ identity for composition and permutations of its products
    coerce:(x:List None) -> %
      ++ [] = 1
    _/:(%,List %) -> List %
      ++ lists of applicators
    apply:(List %,%) -> List %
      ++ lists of applicators

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

    -- Prop (arity)
    dom(f:%):NNI == rep(f).domain
    cod(f:%):NNI == rep(f).codomain
    dat(f:%):T == rep(f).data
    arity(f:%):Prop == dom(f)/cod(f)

    retractIfCan(f:%):Union(K,"failed") ==
      dom(f)=0 and cod(f)=0 => retract(dat f)$T
      return "failed"
    retract(f:%):K ==
      dom(f)=0 and cod(f)=0 => retract(dat f)$T
      error "failed"

    -- basis
    basisVectors():List % == [per [0,1,entries(row(1,i)$SquareMatrix(dim,K))::T] for i in 1..dim]
    basisForms():List % == [per [1,0,entries(row(1,i)$SquareMatrix(dim,K))::T] for i in 1..dim]

    -- manipulation
    map(f:K->K, g:%):% == per [dom g,cod g,unravel(map(f,ravel dat g))$T]
    ravel(g:%):List K == ravel dat g
    unravel(p:Prop,r:List K):% ==
      dim^(dom(p)+cod(p)) ~= #r => error "failed"
      per [dom(p),cod(p),unravel(r)$T]
    tensor(x:%):T == dat(x)

    -- sum
    (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:%):% ==
      dat(f)=0 => g
      dat(g)=0 => f
      dom(f) ~= dom(f) or cod(g) ~= cod(g) => error "arity"
      per [dom f, cod f,dat(f)-dat(g)]

    0 == per [0,0,0]

    -- repeated sum
    (p:NNI * f:%):% ==
      p=1 => f
      q:=subtractIfCan(p,1)
      q case NNI => q*f + f
      per [dom f,cod f,0*dat(f)]

    -- evaluation:
    --
    -- g f : A^n -> A^p = g:A^m -> A^p * f:A^n -> A^m
    apply(gg:%,ff:%):% ==
      -- partial application from the right
      n:=subtractIfCan(dom ff,cod gg)
      f:=ff; n case NNI => f:=1^n*ff
      m:=subtractIfCan(cod gg, dom ff)
      g:=gg; m case NNI => g:=1^m*gg
      f/g

    -- f/g : A^n -> A^p = f:A^n -> A^m / g:A^m -> A^p
    (ff:% / gg:%):% ==
      -- partial application from the left
      n:=subtractIfCan(dom ff,cod gg)
      f:=ff; n case NNI => f:=ff*1^n
      m:=subtractIfCan(cod gg, dom ff)
      g:=gg; m case NNI => g:=gg*1^m
      --dom(g) ~= cod(f) => error "arity"
      r:T := product(dat f, dat g)
      g1:Integer:=dom(f)+1
      f1:Integer:=dom(f)+cod(f)+1
      for i in 0..cod(f)-1 repeat
        r := contract(r,g1,f1-i)
      per [dom(f),cod(g),r]

    -- lists of applicators
    apply(gg:List %,ff:%):List % == [ff/g for g in gg]
    (f:% / g:List %):List % == [f/gg for gg in g]

    -- permutations and identities
    coerce(p:List PositiveInteger):% ==
      r:=per([1,1,kroneckerDelta()$T])^#p
      #p = 1 and p.1 = 1 => return r
      p1:List Integer:=[i for i in 1..#p]
      p2:List Integer:=[#p+i for i in p]
      p3:=concat(p1,p2)
      per [#p,#p,reindex(dat r,p3)]
    coerce(p:List None):% == per [0,0,1]
    1:% == per [0,0,1]
    coerce(x:K):% == 1*x

    -- product
    (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)]

    -- repeated product
    (f:% ^ p:NNI):% ==
      p=1 => f
      q:=subtractIfCan(p,1)
      q case NNI => f^q * f
      per [dom f,cod f,1]

    -- inherited from Ring
    (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)]

    -- constructors
    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

    -- display operators using basis
    coerce(x:%):OutputForm ==
      dom(x)=0 and cod(x)=0 => return dat(x)::OutputForm
      if size()$gen > 0 then
        gens:List OutputForm:=[index(i::PositiveInteger)$gen::OutputForm for i in 1..dim]
      else
        -- default to numeric indices
        gens:List OutputForm:=[i::OutputForm for i in 1..dim]
      -- input basis
      inps:List OutputForm := []
      for i in 1..dom(x) repeat
        empty? inps => inps:=gens
        inps:=concat [[hconcat(inps.k,gens.j) for j in 1..dim] for k in 1..#inps]
      -- output basis
      outs:List OutputForm := []
      for i in 1..cod(x) repeat
        empty? outs => outs:=gens
        outs:=concat [[hconcat(outs.k,gens.j) for j in 1..dim] for k in 1..#outs]
      -- term list
      bases:List OutputForm:=[]
      coeffs:=ravel dat(x)
      if #inps > 0 and #outs > 0 then
        expon:List List OutputForm:=concat([[[i,j] for j in outs] for i in inps])
        bases:=[scripts('e::OutputForm,ij) for ij in expon]
      else if #inps > 0 then
        bases:=[sub('e::OutputForm,i) for i in inps]
      else if #outs > 0 then
        bases:=[super('e::OutputForm,j) for j in outs]
      terms:=[(k=1 => base;k::OutputForm*base) for base in bases for k in coeffs | k~=0]
      empty? terms => return 0::OutputForm
      return reduce(_+,terms)

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1735447310595016784-25px003.spad
      using system compiler.
   LIN abbreviates domain LinearOperator 
------------------------------------------------------------------------
   initializing NRLIB LIN for LinearOperator 
   compiling into NRLIB LIN 
   importing List NonNegativeInteger
************* USER ERROR **********
available signatures for Rep: 
    NONE
NEED Rep: () -> ?
   Semantic Errors: 
      [1]  OrderedFinite is not a known type

****** comp fails at level 1 with expression: ******
((DEF (|Rep|) (NIL)
  (|Record| (|:| |domain| (|NonNegativeInteger|))
            (|:| |codomain| (|NonNegativeInteger|))
            (|:| |data| (|CartesianTensor| 1 |dim| K)))))
****** level 1  ******
x:= (DEF (Rep) (NIL) (Record (: domain (NonNegativeInteger)) (: codomain (NonNegativeInteger)) (: data (CartesianTensor (1) dim K))))
m:= $EmptyMode
f:=
((((|#| #) (< #) (<= #) (= #) ...)))

   >> Apparent user error:
   unspecified error

Getting Started

Consult the source code above for more details.

Basis

dim:=2

(1)

L:=LinearOperator(dim,OVAR [x,y],FRAC POLY INT)

   LinearOperator is an unknown constructor and so is unavailable. Did 
      you mean to use -> but type something different instead?

Conveniences

macro Σ(x,b,i)==reduce(+,[x*b.i for i in 1..dim])

macro /\(x,s)==superscript(x,s)

macro \/(x,s)==subscript(x,s)

Construction

A1:L := Σ(a1\/[i],dx,i)

   L is not a valid type.

Composition (evaluation)

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

(2)

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

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named arity
      with argument type(s) 
                        Fraction(Polynomial(Integer))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Partial Evaluation

BBA1 := B/A1

(3)

BBA2 := B A1

   There are no library operations named B 
      Use HyperDoc Browse or issue
                                 )what op B
      to learn if there is any operation containing " B " in its name.

   Cannot find a definition or applicable library operation named B 
      with argument type(s) 
                                Variable(A1)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

List of Applicators

B/[A1,A2]

   There are 11 exposed and 16 unexposed library operations named / 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op /
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named / 
      with argument type(s) 
                                 Variable(B)
                     List(OrderedVariableList([A1,A2]))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Powers

AB3:=(AB1*AB1)*AB1;

arity(AB3)

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named arity
      with argument type(s) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Sums

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

(4)

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

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named arity
      with argument type(s) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Multiplication

A3s:=(A1+A1)+A1

(5)

arity(A3s)

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named arity
      with argument type(s) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

B3s:=(B1+B1)+B1

(6)

arity(B3s)

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named arity
      with argument type(s) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Product

AB11:=A1*B1

(7)

arity(AB11)

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named arity
      with argument type(s) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Permutations and Identities

H:L:=[1,2]

   L is not a valid type.

Manipulations

tensor AB

   There are 1 exposed and 1 unexposed library operations named tensor 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                             )display op tensor
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      tensor with argument type(s) 
                                Variable(AB)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Examples

Another kind of diagram::

__ _ __ ) = \ / __ )

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

   L is not a valid type.

Handle

          
 _/ _\__ 

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

   L is not a valid type.

Back to the top.

Comments

Please leave comments and suggestions.

Thanks

Bill Page

  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.

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

   There are no library operations named VectorSpace 
      Use HyperDoc Browse or issue
                            )what op VectorSpace
      to learn if there is any operation containing " VectorSpace " in 
      its name.

   Cannot find a definition or applicable library operation named 
      VectorSpace with argument type(s) 
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Unfortunately:

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

   There are no library operations named VectorSpace 
      Use HyperDoc Browse or issue
                            )what op VectorSpace
      to learn if there is any operation containing " VectorSpace " in 
      its name.

   Cannot find a definition or applicable library operation named 
      VectorSpace with argument type(s) 
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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.

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

   There are no library operations named VectorSpace 
      Use HyperDoc Browse or issue
                            )what op VectorSpace
      to learn if there is any operation containing " VectorSpace " in 
      its name.

   Cannot find a definition or applicable library operation named 
      VectorSpace with argument type(s) 
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.