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Re: Partitioned matrix operations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg100664] Re: Partitioned matrix operations
  • From: "Steve Luttrell" <steve at _removemefirst_luttrell.org.uk>
  • Date: Wed, 10 Jun 2009 17:11:21 -0400 (EDT)
  • References: <h0l467$oof$1@smc.vnet.net> <h0nug5$b6e$1@smc.vnet.net>

Your proposed use of Inverse works only if A, B and C commute (where 
appropriate). You can verify that your proposed inverse is not correct by 
premultiplying it by the original matrix thus:

{{A, B}, {C, 0}} . {{0, C^(-1)}, {B^(-1), -A (B C)^(-1)}}

When you expand this out the upper right element is

A C^(-1) - B A (B C)^(-1)

which does not simplify to 0 (as you would have liked) when A, B and C don't 
commute (where appropriate).

Anyway, back to the original problem. Since it is only a 2x2 block matrix 
you can solve it manually thus:

{{A,B},{C,0}} . {{U,V},{W,X}} = {{1,0},{0,1}}

where you have to solve for the matrices U, V, W, X, and the 1 and 0 symbols 
on the r.h.s. stand for the appropriate matrices.

I won't say any more in case this is a homework problem.

-- 

Stephen Luttrell
West Malvern, UK

"dh" <dh at metrohm.com> wrote in message news:h0nug5$b6e$1 at smc.vnet.net...
>
>
> Hi,
>
> you may symbolically invert your matrix:
>
> Inverse[M]
>
> giving:
>
> {{0, 1/C}, {1/B, -(A/(B C))}}
>
> Now it is up to you to ensure that C,B and B C are invertible.
>
> Daniel
>
>
>
>
>
> Joe Hays wrote:
>
>> Hello,
>
>> Here's a Mathematica newbie question. Say I have a matrix, M, defined as,
>
>>
>
>> M = {{A, B}, {C, 0}}
>
>>
>
>> where A is nxn, B is nxm, C is mxn, and the zero sum matrix is mxm. I 
>> would
>
>> like to perform an operation on the matrix M without fully defining A, B,
>
>> and C and get a result in terms of A, B, and C. For example, if I wanted 
>> to
>
>> determine the matrix inverse of M in terms of A, B, and C.
>
>>
>
>> Is this possible in Mathematica? I've unfortunately not found anything in
>
>> the docs that indicates that this is possible.
>
>>
>
>> Thx.
>
>>
>
>>
>
>
> 



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