im_inv inverse image

## Syntax

    [X,dim]=im_inv(A,B [,tol])
                [X,dim,Y]=im_inv(A,B, [,tol])

## Arguments

A,B

  two real or complex matrices with equal number of columns

X

  orthogonal or unitary square matrix of order equal to the number of columns of `A`

dim

  integer (dimension of subspace)

Y

  orthogonal matrix of order equal to the number of rows of `A` and `B`.

## Description

`[X,dim]=im_inv(A,B)` computes `(A^-1)(B)` i.e vectors whose image through `A` are in range(`B`)

The `dim` first columns of `X` span `(A^-1)(B)`

`tol` is a threshold used to test if subspace inclusion; default value is `tol = 100*%eps`. If `Y` is returned, then `[Y*A*X,Y*B]` is partitioned as follows: `[A11,A12;0,A22]`,`[B1;0]`

where `B1` has full row rank (equals `rank(B)`) and `A22` has full column rank and has `dim` columns.

## Examples

    A=[rand(2,5);[zeros(3,4),rand(3,1)]];B=[[1,1;1,1];zeros(3,2)];
    W=rand(5,5);A=W*A;B=W*B;
    [X,dim]=im_inv(A,B)
    svd([A*X(:,1:dim),B])   //vectors A*X(:,1:dim) belong to range(B)
    [X,dim,Y]=im_inv(A,B);[Y*A*X,Y*B]

## See also

- rowcomp
- spaninter
- spanplus
- linsolve