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Assume $A$ is a $n\times n$ matrix and $B=\left[\begin{matrix}B_{11}&B_{12}\\B_{21}&B_{22}\\\end{matrix}\right]$, then do we have $$\|A\otimes B\|=\|\left[\begin{matrix}A\otimes B_{11}&A\otimes B_{12}\\ A\otimes B_{21}&A\otimes B_{22}\end{matrix}\right]\|?$$ where the $\|\cdot\|$ denotes the spectral norm If not, can you give me a counterexample?

My try:$\|A\otimes B\|=\|B\otimes A\|=\|\left[\begin{matrix}B_{11}\otimes A&B_{12}\otimes A\\B_{21}\otimes A&B_{22}\otimes A\end{matrix}\right]\|$, but then I don't know how to do next

Any hints will be appreciated. Hope for your help sincerely

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  • $\begingroup$ I think they are orthogonal similar matrices. So they have the same spectral norm $\endgroup$ – Good boy Nov 5 '18 at 1:31

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