Scilab Function
Last update : 12/01/2004

im_inv - inverse image

### Calling Sequence

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

### Parameters

- `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`,

### Author

F. Delebecque INRIA