Suppose I have a $m\times n$ matrix $\mathbf M$, and a unit vector $\hat v$, of dimension $n$. What restrictions do I need to apply to $\mathbf M$ so that $\lVert \mathbf M\cdot \hat v\lVert \leq 1$ for any $\hat v$?
2026-03-29 04:48:20.1774759700
Restrictions on a Matrix-Vector product
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This is the same as saying that the "operator norm" of $M$ is less than $1$, which is equivalent to the largest singular value of $M$ being $1$. That is, the largest eigenvalue of $M^*M$ has to be less than one. What kind of conditions are you looking for?