General matrix inverses

• To: mathgroup at smc.vnet.net
• Subject: [mg23516] General matrix inverses
• From: bt585 at freenet.carleton.ca (Michael Chang)
• Date: Tue, 16 May 2000 02:45:02 -0400 (EDT)
• Organization: The National Capital FreeNet
• Sender: owner-wri-mathgroup at wolfram.com

```Hi everyone,

I have a question about how one might use Mathematica 3.0.x to perform
general matrix inverses for square matrices.

Specifically, suppose that I have a square complex matrix A having
dimensions (m+n)x(m+n).  Let A be partitioned as (please bear with me in
the following ascii representation)

A = [  A11   A12
0     A22]

where A11, A12, A22 are complex matrices of dimensions (nxn), (nxm), and
(mxm), respectively.  (The zero is a zero matrix having dimensions (mxn).)

Is there a way that I can represent the above generalization in
Mathematica without explicitly defining dimensions n or m?  If so, how can
I get Mathematica to produce

A^(-1) = [A11^(-1)   -A11^(-1)A12*A22^(-1)
0                 A22^(-1)]

assuming, of course, that inverses for A11 and A22 exist?

Moreover, can Mathematica 3.0.x recognize and utilize many well known
(finite dimensional) matrix identities (e.g. matrix inversion lemma) for
general matrix simplification, where, generally, the size of individual
matrices is left unspecified (but all matrix dimensions are assumed to be
compatable/consistent), and the invertability of certain matrices is
assumed?  (Since this is all symbolic, I am hoping that this is the case
...) Or is this something that Mathematica 4.x can do?