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# 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.

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