Re: Block-defined matrices

• To: mathgroup at smc.vnet.net
• Subject: [mg16005] Re: Block-defined matrices
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Sat, 20 Feb 1999 02:52:00 -0500
• Organization: University of Western Australia
• References: <7ag4sj\$b1d@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Roberto Pratolongo wrote:

> I've a problem of matrix algebra. I want to commonly manage
> matrices(calculate their inverse,determinant,etc.): they are symbolically
> defined by square blocks.
> For example, let  M ={{A,B},{C,D}}, where A,B,C,D are 3x3 blocks.
>
> So, it exists a way to obtain the output of e.g. Inverse[M] described in
> terms of
> A, Inverse[A], B, Inverse[B], C, Inverse[C], D, Inverse[D] ?

It is not too hard to show (by hand) that, if A,B,C,D are invertible,
the inverse can be written in the form,

{{Inverse[A - B.Inverse[D].C], Inverse[C - D.Inverse[B].A]},
{Inverse[B - A.Inverse[C].D], Inverse[D - C.Inverse[A].B]}}

If, e.g., D is not invertible then the [[1,1]] entry of the inverse can
be replaced by

-Inverse[C].D.Inverse[B-A.Inverse[C].D]

etc.

I am not aware of any general Mathematica tools for performing such
operations though ...

Cheers,
Paul

____________________________________________________________________
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia
Nedlands WA  6907                     mailto:paul at physics.uwa.edu.au
AUSTRALIA                        http://www.physics.uwa.edu.au/~paul

God IS a weakly left-handed dice player
____________________________________________________________________

```

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