Given matrices $A_{n\times n}, B_{n\times m}, C_{m\times m}$ such that $A^iBC^{N-i}$ is matrix with all zeros except upper right element for all $i$ from $0$ to $N$, what can we say about Frobenius norm of sum $\left\|\sum_{i=0}^NA^iBC^{N-i}\right\|$?
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3$\begingroup$ Surely you can't say anything? Multiplying $B$ by a non-zero scalar doesn't change the zero pattern of $A^i B C^{N-i}$, but will let you change the Frobenius norm to whatever you want. $\endgroup$– Nathaniel JohnstonCommented Sep 4, 2023 at 17:22
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$\begingroup$ @NathanielJohnston If $B$ and $Q$ are $n\times n$ matrices, with $Q$ orthogonal, what can we say about the Frobenius norm $\|QB\|_F$? $\endgroup$– Federico PoloniCommented Sep 4, 2023 at 22:11
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$\begingroup$ @NathanielJohnston Unfortunately, all I can say is that $\left\|\sum_{i=0}^NA^iBC^{N-i}\right\| = \sum_{i=0}^N\left\|A^iBC^{N-i}\right\| $. I wonder if we can formulate some other properties in this case. $\endgroup$– D.UltCommented Sep 5, 2023 at 6:07
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