As you know, we have the two following inequalities for sum and product (for Frobenius norm):
$\|A+B\|_F\leq\|A\|_F+\|B\|_F$
and
$\|AB\|_F\leq\|A\|_F\|B\|_F$.
The question is, under which conditions, the inequalities both turn to equality.
2026-02-23 04:49:47.1771822187
matrix F-norm inequalities on matrix sum and product
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Hint. The Frobenius norm of a matrix is just the $2$-norm of its vectorization.