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Block matrix transformation of specific form

by SNicolas
Tags: block, form, matrix, specific, transformation
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Feb20-13, 04:38 AM
P: 2
Hi everyone,

I am trying to solve the following problem. Is there exist a transformation matrix T, different then the block diagonal, with all blocks the same, such that the form of the matrix A=[A1 A2 ; I 0], is preserved? All blocks of A are in R^{nxn}, I is identity and 0 is zero matrix. In other words, is there exist matrix T (again, not block diagonal with all blocks the same) such that T^{-1}*A*T=B, where B=[B1 B2 ; I 0]?
By intuition, such matrix T does not exist, but I do not know how this can be shown.
If anyone has an idea about this please help. Thank you in advance.

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Feb20-13, 08:17 AM
P: 12,081
So T should be like [t t; t t]?
For the lower blocks of B, this would give the equations 1*t + 0*t = 1 and 1*t + 0*t = 0 => contradiction
Feb20-13, 08:31 AM
P: 2
Thanks for your reply.

Since I am looking for the existence of T, in general it could be
T=[T11 T12 ; T21 T22]. I know that T=[T 0; 0 T] holds, but I want to show that this is the only case. For example T=[T11 0 ; 0 T22] cannot hold, neither the example that you proposed.

Feb20-13, 11:23 AM
P: 12,081
Block matrix transformation of specific form

Ok, I was not sure what "with all blocks the same" means.
Anyway, [t t; t t] matrix would not have an inverse matrix.

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