Norm of rows of a matrix with a given spectral norm

Question

Given a $n \times n$ matrix $M$ such that $\|M\|_2 = c$ (where $\|\|_2$ denotes spectral norm, or operator norm), is it true that for all $i = 1...n$ it holds that $$\sqrt{(\sum_{k=0}^n M(i,k)^2)} \leq c$$ Or, does the norm (vector $l_2$ norm) of the rows of $M$ is bounded by its spectral norm? I believe so, but still not sure, will be glad for intuition explanation (or any hint).

Answer 1

The same with the columns amounts to $\lVert Ae_i\lVert_2\leq \lVert A\lVert_2 \lVert e_i\lVert_2$. But this simply follows from the fact that the spectral norm is defined by

$$\lVert A\lVert_2=\sup_{u\neq0}\frac{\lVert Au\lVert_2}{\lVert u\lVert_2}$$

For the rows, simply use $A^T$ instead.

See

• https://en.wikipedia.org/wiki/Matrix_norm
• http://mathworld.wolfram.com/SpectralNorm.html

And also have a look at this MSE question for a proof that $\lVert A \lVert_2$ is indeed the spectral radius: Norm of a symmetric matrix equals spectral radius