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Consider a possibly-rectangular (only the rectangular case is of real interest) matrix $M$, and two diagonal matrices $D_1, D_2$.

We want $$D_1 M+M D_2=0$$ Clearly $D_1=0,D_2=0$ is sufficient.

But is is not always necessary:

  • if $M=0$, any $D_1, D_2$ will do.
  • If $M$ is diagonal, $D_1=-D_2$ suffices.

What conditions must $M$ obey for the trivial solution $D_1=0,D_2=0$ to be the only solution? Does it suffice that $M$ does not commute with diagonal matrices?

Wouter
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1 Answers1

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There are always infinitely many solutions. Suppose $M$ is a $k\times n$ matrix. Choose $a\in\mathbb{R}$, let $I$ denote the identity matrix and let $D_1 = a I_{k}$ and let $D_2 = -a I_{n}$. Then we get $$ D_1M + MD_2 = aI_kM + M(-a)I_n = aI_kM-aMI_n = aM - aM = 0. $$

Marc
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