Block matrix transformation of specific form

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Discussion Overview

The discussion revolves around the existence of a transformation matrix T that preserves the structure of a specific block matrix A when transformed into another block matrix B. The matrices A and B have a defined form, and the inquiry focuses on whether T can be non-block diagonal while maintaining this structure.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Nicolas proposes the problem of finding a transformation matrix T that is not block diagonal and has all blocks the same, questioning whether such a matrix can exist to transform A into B.
  • One participant suggests that T should take the form [t t; t t], leading to a contradiction in the equations derived from the lower blocks of B.
  • Nicolas clarifies that T could generally be represented as [T11 T12; T21 T22] and emphasizes the need to show that only the block diagonal case holds.
  • Another participant expresses uncertainty about the meaning of "with all blocks the same" and notes that the proposed matrix [t t; t t] would not have an inverse.

Areas of Agreement / Disagreement

Participants express differing views on the form of the transformation matrix T and its implications, with no consensus reached on the existence of such a matrix that is not block diagonal.

Contextual Notes

The discussion includes assumptions about the structure and properties of the transformation matrix T, as well as the implications of its form on the invertibility and transformation of the block matrices A and B.

SNicolas
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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
 
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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
 
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.
 
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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