I have the Frobenius norm of two products, $\lVert AB\rVert_F$ and $\lVert AC\rVert_F$. $A$, $B$, and $C$ are matrices, the dimensions do not matter as long as they are compatible and $B$ and $C$ have the same size. I know that $\lVert B\rVert \leq \lVert C\rVert$, is there any inequality I can obtain with the original two norms (e.g. $\lVert AB\rVert_F \leq \lVert AC\rVert_F$, or anything else)?
2026-03-26 04:30:08.1774499408
Frobenius norm product with two inequalities
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Let $A=B=\pmatrix{1&0\\ 0&0}$ and $C=\pmatrix{0&0\\ 0&2}$. Then $\|B\|_F=1<2=\|C\|_F$ but $\|AB\|_F=1>\|AC\|_F=0$.