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Let $A_1,\dots,A_n$ be matrices, with no row or column of $0s$, and such that for every $i=1,\dots,n$ there does not exist a decomposition of $A_i$ of the form $$ A_i = \oplus_{j=1}^n B_j \qquad (\exists j_1\neq j_2)\;(\exists k \in \mathbb{R}) \,B_{j_1}=kB_{j_2}, $$ where the direct sum of matrices is defined here.

Is there a reasonable compatibility criterion on the $A_i$ such that $A= \oplus_{i=1}^nA_i$ also does not admit such a decomposition; i.e.: there do not exist $C_1,\dots,C_t$ such that $$ A = \oplus_{i=1}^t C_i \mbox{ and } (\exists t_1\neq t_2)\;(\exists k \in \mathbb{R}) \,C_{i_1}=kC_{i_2}. $$


Non-example: An interesting non-example, which illustrates part of the issue is this: $$ A_1\triangleq \begin{pmatrix} 1 & 1 & 0\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} A_2 \triangleq (2), $$ then $A_1,A_2$ are distinct and of distinct size but, $B_1$ is the 2x2 matrix of 1s, $B_2=(1)$, $B_3=A_2$ gives the decomposition I don't want.

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    $\begingroup$ What is a direct sum of matrices? $\endgroup$ – abx Feb 12 at 11:34
  • $\begingroup$ I am not sure I understand correctly what you mean with 'decomposition', but what happens if all the $A_i$ are equal, for instance? $\endgroup$ – Federico Poloni Feb 12 at 11:43
  • $\begingroup$ @abx I added in a link and a case where things break. I'm curious what a reasonable condition on the $A_i$ are... $\endgroup$ – AIM_BLB Feb 12 at 12:32

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