Is it true that $\Vert Mv\Vert_{2}\leq \Vert M\Vert_{2}\Vert v\Vert_{2}$, when $M\in\mathbb{R}^{3\times3}$, and $v\in\mathbb{R}^3$?

Question

Is it true that $\Vert Mv\Vert_{2}\leq \Vert M\Vert_{2}\Vert v\Vert_{2}$, when $M\in\mathbb{R}^{3\times3}$, and $v\in\mathbb{R}^3$, where $\Vert\cdot\Vert_{2}$ is the $\ell_{2}$ norm, and when applied to a matrix, the matrix norm induced by vector norms is used.

I know it is true when both arguments are vectors, but I have no knowledge of matrix norms.

I think a better answer would be one that is true for arbitrary dimensions, but I gave my particular dimensions since the matrix works out to be square, which may be important.

Any help with getting a clearer understanding of this is greatly appreciated.

Thanks

Answer 1

This is not specific to the euclidean norm. If you define a matrix norm through the expression: $$ \|M\| = \sup_{u \ne 0}\dfrac{\|M u\|}{\|u\|} $$

then, for any $v \ne 0$, you certainly have that $\|M\| \ge \dfrac{\|Mv\|}{\|v\|}$, i.e. $\|Mv\| \leq \|M\| \|v\|$ (the inequality is trivially satisfied wghen $v=0$). So, the inequality holds true for any matrix norm that is induced by a vector norm (this is not always the case).