Is the spectral norm of a matrix a strictly convex function?

Question

Let $\|A\|_2=\sigma_1(A)$ denote the spectral norm of matrix $A$, i.e., its maximum singular value.

It is well known that any norm is convex. Is the spectral norm strictly convex?

Answer 1

As a function, a norm is never strictly convex, due to homogeneity. But there is a useful concept of strictly convex norm which is different from "the norm is a strictly convex function". Namely, a norm is strictly convex if $$\|a+b\|<\|a\|+\|b\|\quad \text{whenever $a, b$ are linearly independent}$$

This is equivalent to saying that the restriction of the norm to any line that does not pass through the origin is a strictly convex function.

That said, the spectral norm is not strictly convex. Any norm that has "maximum" in its definition is unlikely to be strictly convex, because the maximum can stay the same while smaller elements are perturbed. For example, let $$A = \begin{pmatrix}1 & 0 \\ 0 & 1/2\end{pmatrix}, \quad B = \begin{pmatrix}1 & 0 \\ 0 & 1/3\end{pmatrix} $$ then $\|A\|=\|B\|=1$ and $\|A+B\|=2$, although $A$ and $B$ are linearly independent (neither is a multiple of the other).