# Condition for non-existence of trivial matrix decomposition

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.

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