Assume $A \in \mathbb{R}^{n\times m}$ is a $n\times m$ matrix, $B\in \mathbb{R}^{m\times n}$ is a $m\times n$ matrix, $\|AB\|_F \neq \|BA\|_F$ is definitely true at the most cases, but is there any rate between $\|AB\|_F$ and $\|BA\|_F$?
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The rate would be related to $n, m$ – Math Stat Jul 06 '23 at 07:26
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The spectral radius of $AB$ equals that of $BA$, so you could say that $$\large{ \lim_{k\to\infty} \bigg|(AB)^k\bigg|F^{\frac1k} = \lim{k\to\infty} \bigg|(BA)^k\bigg|_F^{\frac1k} \ }$$ – greg Jul 08 '23 at 17:23
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They can be arbitrarily different. See this answer. Take $$ A = \begin{pmatrix} x & 1 \\ 0 & 0 \end{pmatrix} \qquad B = \begin{pmatrix} 0 & 0 \\ 1 & y \end{pmatrix} $$ for any $x,y \in \mathbb{R}$. Then $$ \|AB\|_F = \sqrt{1+y^2} \qquad \|BA\|_F = \sqrt{1+x^2}. $$ So I don't see any relationship between the two without further constraints.
Rammus
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If $A$ and $B$ are random matrix, for example, with iid entries, does the rate can be computed in probability? – Math Stat Jul 06 '23 at 16:25
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