Relationship between the Frobenius Norm of AB and BA?

Question

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 $\|BA\|_F$?

Answer 1

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.