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<Chapter Label="Vector and Matrix Objects">
<Heading>Vector and Matrix Objects</Heading>

This chapter describes an interface to vector and matrix objects
which are not represented by plain lists (of plain lists),
cf. Chapters <Ref Chap="Row Vectors"/> and <Ref Chap="Matrices"/>.


<Section Label="Concepts and Rules for Vector and Matrix Objects">
<Heading>Concepts and Rules for Vector and Matrix Objects</Heading>

<#Include Label="MatObj_Overview">

</Section>


<Section Label="Categories of Vector and Matrix Objects">
<Heading>Categories of Vector and Matrix Objects</Heading>

Currently the following categories of vector and matrix objects
are supported in &GAP;.
More can be added as soon as there is need for them.


<#Include Label="IsVectorObj">
<#Include Label="IsMatrixObj">
<#Include Label="IsMatrixOrMatrixObj">
<#Include Label="IsRowListMatrix">

</Section>


<Section Label="Defining Attributes of Vector and Matrix Objects">
<Heading>Defining Attributes of Vector and Matrix Objects</Heading>

<#Include Label="BaseDomain">
<#Include Label="ConstructingFilter">
<#Include Label="CompatibleVectorFilter">
<#Include Label="Length_IsVectorObj">
<#Include Label="NumberRowsNumberColumns">

</Section>


<Section Label="Constructing Vector and Matrix Objects">
<Heading>Constructing Vector and Matrix Objects</Heading>

<#Include Label="NewVector">
<#Include Label="Vector">
<#Include Label="VectorObj_ZeroVector">
<#Include Label="NewMatrix">
<#Include Label="MatObj_Matrix">
<#Include Label="MatObj_ZeroMatrix">
<#Include Label="MatObj_IdentityMatrix">

</Section>


<Section Label="Operations for Base Domains of Vector and Matrix Objects">
<Heading>Operations for Base Domains of Vector and Matrix Objects</Heading>

<#Include Label="OneOfBaseDomain">

</Section>


<Section Label="Operations for Vector and Matrix Objects">
<Heading>Operations for Vector and Matrix Objects</Heading>

<#Include Label="MatrixObjCompare">
<#Include Label="Unpack">
<#Include Label="ChangedBaseDomain">
<#Include Label="Randomize">

</Section>


<Section Label="List Like Operations for Vector Objects">
<Heading>List Like Operations for Vector Objects</Heading>

The following operations that are defined for lists are useful
also for vector objects.
(More such operations can be added if this is appropriate.)

<#Include Label="ElementAccessVectorObj">
<#Include Label="MatObj_PositionNonZero">
<#Include Label="MatObj_PositionLastNonZero">
<#Include Label="MatObj_ListOp">

</Section>


<Section Label="Arithmetical Operations for Vector Objects">
<Heading>Arithmetical Operations for Vector Objects</Heading>

<#Include Label="VectorObj_UnaryArithmetics">
<#Include Label="VectorObj_BinaryArithmetics">
<#Include Label="MatObj_AddVector">
<#Include Label="MatObj_MultVectorLeft">

</Section>


<Section Label="Operations for Vector Objects">
<Heading>Operations for Vector Objects</Heading>

<#Include Label="MatObj_ConcatenationOfVectors">
<#Include Label="MatObj_ExtractSubVector">
<#Include Label="CopySubVector">
<#Include Label="MatObj_WeightOfVector">
<#Include Label="MatObj_DistanceOfVectors">

</Section>


<Section Label="Arithmetical Operations for Matrix Objects">
<Heading>Arithmetical Operations for Matrix Objects</Heading>

<#Include Label="MatrixObj_UnaryArithmetics">
<#Include Label="MatrixObj_BinaryArithmetics">

</Section>


<Section Label="Operations for Matrix Objects">
<Heading>Operations for Matrix Objects</Heading>

<#Include Label="MatObj_MatElm">
<#Include Label="MatObj_SetMatElm">
<#Include Label="ExtractSubMatrix">
<#Include Label="MutableCopyMatrix">
<#Include Label="CopySubMatrix">
<#Include Label="CompatibleVector">
<#Include Label="RowsOfMatrix">
<#Include Label="MatObj_CompanionMatrix">

</Section>


<Section Label="Operations for Row List Matrix Objects">
<Heading>Operations for Row List Matrix Objects</Heading>

In general, matrix objects are not lists in the sense of <Ref Filt="IsList"/>,
and they need not behave like lists,
that is, they need not obey all the rules for lists that are stated in
Chapter <Ref Chap="Lists"/>.
There are situations where one wants to have matrix objects
that can on the one hand benefit from &GAP;'s method selection,
as is explained in
Section <Ref Sect="Concepts and Rules for Vector and Matrix Objects"/>,
and do on the other hands support access to &GAP; objects that represent
their rows (which are suitable vector objects).
Matrix objects whose
<Ref Attr="ConstructingFilter" Label="for a matrix object"/> value implies
<Ref Filt="IsRowListMatrix"/> support the operations described in this
section.

<P/>

One implementation of such matrices is given by the
<Ref Attr="ConstructingFilter" Label="for a matrix object"/> value
<Ref Filt="IsPlistMatrixRep"/>,
and any row of these matrices is a vector object in
<Ref Filt="IsPlistVectorRep"/>.
Note that these objects do <E>not</E> lie in <Ref Filt="IsList"/>
(and in particular not in <Ref Filt="IsPlistRep"/>),
thus we are allowed to define the above operations only restrictively,
as follows.

<P/>

Unbinding an entry in a row or unbinding a row in a matrix is allowed only
in the last position,
that is, the vector and matrix objects insist on being dense.
All rows of a matrix must have the same length and the same base domain.

<#Include Label="RowListMatObj_[]">
<#Include Label="RowListMatObj_[]_ASS">
<#Include Label="RowListMatObj_{}">
<#Include Label="RowListMatObj_{}_ASS">
<#Include Label="RowListMatObj_IsBound">
<#Include Label="RowListMatObj_Unbind">
<#Include Label="RowListMatObj_Add">
<#Include Label="RowListMatObj_Remove">
<#Include Label="RowListMatObj_Append">
<#Include Label="RowListMatObj_ShallowCopy">
<#Include Label="RowListMatObj_ListOp">

</Section>


<Section Label="Basic operations for row/column reductions">
<Heading>Basic operations for row/column reductions</Heading>

<#Include Label="MultMatrixRow">
<#Include Label="MultMatrixRowRight">
<#Include Label="MultMatrixColumn">
<#Include Label="MultMatrixColumnLeft">
<#Include Label="AddMatrixRows">
<#Include Label="AddMatrixRowsRight">
<#Include Label="AddMatrixColumns">
<#Include Label="AddMatrixColumnsLeft">
<#Include Label="SwapMatrixRows">
<#Include Label="SwapMatrixColumns">

</Section>


<Section Label="Implementing New Vector and Matrix Objects Types">
<Heading>Implementing New Vector and Matrix Objects Types</Heading>

The first step in the design of a new type of vector or matrix objects
is to create a new filter that serves as the
<Ref Attr="ConstructingFilter" Label="for a vector object"/>
of the new objects,
see the sections <Ref Sect="Available Representations of Vector Objects"/>
and <Ref Sect="Available Representations of Matrix Objects"/>
for an overview of such filters that are already available.

<P/>

Here we list those operations for vector and matrix objects
for which no default methods can be installed.
When one implements a new type of vector or matrix objects
then one has to install specific methods at least for these operations,
in order to make the objects behave as described in this chapter.
It is advisable to install specific methods also for other
operations, for performance reasons.
The installations of default methods can be found in the file
<F>lib/matobj.gi</F> of the &GAP; distribution.
There one can check for which operations it makes sense to overload them
for the new type of vector or matrix objects.
Note that the specific methods must be installed with
<Ref Func="InstallTagBasedMethod"/> whenever the default method is installed
with this function.

<P/>

<E>Vector objects</E>

<List>
<Item>
  <Ref Attr="BaseDomain" Label="for a vector object"/>,
</Item>
<Item>
  <Ref Attr="Length" Label="for a vector object"/>,
</Item>
<Item>
  <Ref Oper="\[\]" Label="for a vector object and an integer"/>,
</Item>
<Item>
  <Ref Oper="\[\]\:\=" Label="for a vector object and an integer"/>
  (with consistency checks if the global option <C>check</C> is not set to
  <K>false</K>),
</Item>
<Item>
  <Ref Oper="\<"/> (see <Ref Oper="\<" Label="for two vector objects"/>),
</Item>
<Item>
  <Ref Attr="ConstructingFilter" Label="for a vector object"/>,
</Item>
<Item>
  <Ref Oper="NewVector"/>
  (with consistency checks if the global option <C>check</C> is not set to
  <K>false</K>, install the method with <Ref Func="InstallTagBasedMethod"/>).
</Item>
</List>

<P/>

<E>Matrix objects</E>

<List>
<Item>
  <Ref Attr="BaseDomain" Label="for a matrix object"/>,
</Item>
<Item>
  <Ref Attr="NumberRows" Label="for a matrix object"/>,
</Item>
<Item>
  <Ref Attr="NumberColumns" Label="for a matrix object"/>,
</Item>
<Item>
  <Ref Oper="MatElm"/>,
</Item>
<Item>
  <Ref Oper="SetMatElm"/>
  (with consistency checks if the global option <C>check</C> is not set to
  <K>false</K>),
</Item>
<Item>
  <Ref Oper="\<"/> (see <Ref Oper="\<" Label="for two matrix objects"/>),
</Item>
<Item>
  <Ref Attr="ConstructingFilter" Label="for a matrix object"/>,
</Item>
<Item>
  <Ref Attr="CompatibleVectorFilter" Label="for a matrix object"/>,
</Item>
<Item>
  <Ref Oper="NewMatrix"/>
  (with consistency checks if the global option <C>check</C> is not set to
  <K>false</K>, install the method with <Ref Func="InstallTagBasedMethod"/>).
</Item>
</List>

<P/>

Methods for <Ref Oper="NewVector"/> and
<Ref Oper="NewMatrix"/> must check their arguments for consistency
(do the given filter and base domain fit together,
are the entries compatible with the given base domain,
is the number of matrix entries a multiple of the given number of columns)
except if the global option <C>check</C> is set to <K>false</K>.
(See Chapter <Ref Chap="Options Stack"/> for information about global
options.)
The same holds for methods for operations that modify mutable vector
or matrix objects,
such as <Ref Oper="\[\]\:\=" Label="for a vector object and an integer"/>,
<Ref Oper="SetMatElm"/>, <Ref Oper="CopySubVector"/>,
<Ref Oper="CopySubMatrix"/>,
and for those methods of
<Ref Oper="Vector" Label="for filter, base domain, and list"/> and
<Ref Oper="Matrix" Label="for filter, base domain, list, ncols"/>
that do not delegate to <Ref Oper="NewVector"/> and
<Ref Oper="NewMatrix"/>, respectively.

</Section>


<Section Label="Available Representations of Vector Objects">
<Heading>Available Representations of Vector Objects</Heading>

The following filters define vector objects for which the the functionality
described in this chapter is supported.

<#Include Label="IsGF2VectorRep">
<#Include Label="Is8BitVectorRep">
<#Include Label="IsPlistVectorRep">
<#Include Label="IsZmodnZVectorRep">

</Section>


<Section Label="Available Representations of Matrix Objects">
<Heading>Available Representations of Matrix Objects</Heading>

The following filters define matrix objects for which the the functionality
described in this chapter is supported.

<#Include Label="IsGF2MatrixRep">
<#Include Label="Is8BitMatrixRep">
<#Include Label="IsPlistMatrixRep">
<#Include Label="IsZmodnZMatrixRep">

</Section>

</Chapter>

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