Is it true that $ \| A \otimes B \| = \|A\|\|B\| $ for any matrix norm $ \|\cdot \| $? If not, does this identity hold for matrix norms induced by $ \ell_p $ vector norms?
2026-03-26 12:35:56.1774528556
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Matrix norm of Kronecker product
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On page 149 exercise 6 in book: Matrix analysis for scientists and engineers, this is true for operator norm. You can see chapter 13 of the book by the link: http://www.siam.org/books/textbooks/OT91sample.pdf
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Theorem 8 here provides the answer: http://www.ams.org/journals/mcom/1972-26-118/S0025-5718-1972-0305099-X/S0025-5718-1972-0305099-X.pdf. As discussed on page 413, the identity holds for all matrix norms induced by $ \ell_p $ vector norms. In fact, it seems to hold for any induced vector norm, or any submultiplicative norm.