# MatrixViews

GNU Free Documentation License, Version 1.2

CoCoALib Documentation Index

## User documentation for MatrixViews

A `MatrixView` offers a means to view one or more existing objects as though they were a `matrix`:

• if you change the entries in the objects then the `MatrixView` changes;
• if you change the entries in the `MatrixView` then the objects change;
• if you destroy or change the structure of the objects then the `MatrixView` may become invalid (and using it could lead to the dreaded undefined behaviour, i.e. probably a crash).

### Examples

NB The views do not make copies, so be careful with temporaries! Look at these examples (`val` is a `RingElem`):

```    // OK
const vector<RingElem> v(3, val);
MatrixView MV = RowMat(v);  // MV reads/writes in the vector v

// NO   this compiles, but the vector disappears after the ";"!!
ConstMatrixView MVGhost = RowMat(vector<RingElem>(3, val));

// OK   NewDenseMat makes a copy of the vector before it disappears
matrix M = NewDenseMat(RowMat(vector<RingElem>(3, val)));
```

### Pseudo-constructors

#### Constant MatrixViews

NB no entry is writable

• `ZeroMat(R, r, c)` -- constant `r`-by-`c` zero matrix over `R`
• `IdentityMat(R, n)` -- constant `n`-by-`n` identity matrix over `R`
• `FilledMat(R, r, c, val)` -- constant `r`-by-`c` matrix over `R` filled with val (a `RingElem`, long, `BigInt`, or `BigRat`)

#### MatrixViews of a vector

You can view a `std::vector<RingElem>`, all of whose entries belong to the same `ring`, as a matrix in three ways:

• `ColMat(v)` -- view a `vector<RingElem>` `v` as a column matrix
• `RowMat(v)` -- view a `vector<RingElem>` `v` as a row matrix
• `DiagMat(v)` -- view a `vector<RingElem>` `v` as a diagonal matrix (NB: only the diagonal entries are writable)

#### MatrixViews of a matrix

• `transpose(M)` -- transposed view of the matrix `M`
• `submat(M, rows, cols)`-- submatrix view into `M`; the rows and columns visible in the submatrix are those specified in the arguments `rows` and `cols` (which are of type `std::vector`)

#### MatrixViews of more matrices

The following pseudo-constructors assemble several matrices into a bigger one; the argument matrices must all have the same `BaseRing`.

• `ConcatVer(A, B)` -- matrix view with the rows of `A` above those of `B`
 A B
• `ConcatHor(A, B)` -- matrix view with the cols of `A` before those of `B`
 A B
• `ConcatDiag(A,B)` -- block diagonal matrix view
 A 0 0 B
• `ConcatAntiDiag(A,B)` -- block antidiagonal matrix view
 0 A B 0
• `BlockMat(A, B, C, D)` -- block matrix view
 A B C D

NB the boundaries of the four submatrices must be aligned.

## Maintainer documentation for MatrixViews

Most of the implementations are quite straightforward; the tricky part was getting the design of the abstract classes right (well, I hope it is right now). Below are a few comments on some less obvious aspects of the implementations.

Note: it is a mathematical fact that the determinant of the 0x0 matrix is 1.

`ZeroMatImpl` and `IdentityMatImpl` are both derived from `MatrixViewBase` rather than `ConstMatrixViewBase` as one might naturally expect. The main reason for this is to simplify the implementation of `BlockMat` views. I wanted to be lazy and implement `ConcatDiag` and `ConcatAntidiag` using `BlockMat`; while this may not be the best implementation, it is a natural approach and should certainly work as one might reasonably expect. However, the pseudo-ctor `BlockMat` has just two signatures: if any one of the submatrices is const then whole result becomes const. I didn't want to implement sixteen different signatures for `BlockMat`, and the easy way out seemed to be to make `ZeroMatImpl` and `IdentityMatImpl` non-const. As a consequence there are a number of useless member functions in `ZeroMatImpl` and `IdentityMatImpl`. I believe this compromise is reasonable. It seemed reasonable to allow `ZeroMatImpl::myAssignZero` to succeed.

There is a small problem with creating a matrix from an empty `std::vector` because there is no indication of what the base ring should be. I have chosen to throw an error if one tries to create a matrix view from an empty vector (in `RowMat`, `ColMat` and `DiagMat`).

The routines which access the (i,j) entry in a BlockMat are messy. I could not see an elegant way to make them simpler (or to avoid repeating similar structure in several places in the code). See Bugs about implementing `BlockMat` in terms of `ConcatVer` and `ConcatHor`.

## Bugs, Shortcomings and other ideas

There is an appalling amount of code duplication in the implementations. I do not yet see a good way of reducing this. I hope someone will sooner or later find an elegant way to avoid the duplication. Maybe a diagonal abstract class for ZeroMatImpl, IdentityMatImpl, DiagMatImpl, ConstDiagMatImpl?

It is a great nuisance to have to implement two very similar classes: one for the const case, and the other for the non-const case. Is there a better way?

Add `ColMat`, `RowMat` and `DiagMat` for a free module element?

Should `submatrix` allow repeated row/col indices? It could lead to some some funny behaviour (e.g. setting one entry may change other entries), so perhaps it would be better to forbid it? Currently, it is forbidden.

The pseudo-ctor for `submatrix` ought to accept begin/end iterators instead of insisting that the caller put the indices in `std::vectors`.

Should there be a more general version of `BlockMat` which allows non-aligned borders? `BlockMat` could be eliminated and replaced by suitable calls to `ConcatVer` and `ConcatHor`.

Tensor product of two matrices: we implement it as a DenseMatrix instead of MatrixView because the latter would give no practical advantage and hide the cost of accessing the entries.

## Main changes

2011

• February (v0.9943):
• optimized implementations for `IsSymmetric`, `IsAntiSymmetric`, `IsDiagonal`, `operator==`
• added `FilledMat` -