Block matrix transformation of specific form

[SNicolas]

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.

Nicolas

 

[mfb]

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

 

[SNicolas]

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.

 

[mfb]

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

 

[blue_raver22]

,

Thank you for sharing your problem. This is an interesting question and one that requires some mathematical analysis to answer.

Based on the given information, it seems that the transformation matrix T would need to have a specific form in order to preserve the structure of A. One possibility could be a block matrix with diagonal blocks and off-diagonal blocks that are all the same. However, this would still result in a block diagonal matrix after the transformation, which is not the desired outcome.

Another possibility could be a transformation matrix with off-diagonal blocks that are not all the same, but still have some kind of pattern or symmetry that would preserve the structure of A. This would require further analysis to determine if such a matrix exists.

In general, it is not always possible to find a transformation matrix that will preserve the structure of a given matrix. This is because the structure of a matrix is determined by its entries, and the transformation matrix can only manipulate these entries in certain ways. Without knowing more about the specific problem and the properties of A, it is difficult to determine if such a matrix T exists.

I would suggest exploring different transformation matrices and their effects on A, and also considering the specific properties and constraints of A to see if a solution can be found. I hope this helps and good luck with your problem.