Convex combination of orthogonal matrices

Question

How would I show that the convex combination of orthogonal matrixes has spectral norm $ \leq 1$? (I have some idea how to do it ... but right now I'm stuck). Also, how would I prove that the unit spectral norm ball is a convex set?

Answer 1

If $A$ is orthogonal, then $\|A\| = 1$ (because $\|Ax\| = \|x\|$ for all $x$).

Suppose $A,B$ are orthogonal and $\mu \in [0,1]$.

Then $\|\mu A + (1-\mu)B \| \le \|\mu A \| + \| (1-\mu)B \| = \mu \|A\| + (1-\mu) \|B\|= 1 + (1-\mu) = 1$.

Any norm ball is convex because the function $x \mapsto \|x\|$ is convex, and if we let $\phi(x) = \|x\|$, then we see that the unit ball is given by $\phi^{-1} (-\infty,1]$, and hence convex (since the level sets of a convex function are convex).

Alternatively, you could note that if $\|A\| \le 1$, $\|B\| \le 1$, then the same formula above (with the second last $=$ replaced by $\le$) shows that $\|\mu A + (1-\mu)B \| \le 1$.