My apologies for not posting in a few days; I’ve been feeling a bit under the weather. Nevertheless, I thought that I would wrap up the work-week with a brief post on the connection between convexity and a very nice class of matrices called positive-definite matrices. Recall that an real matrix is positive-definite if for all , with equality if and only if (here, denotes the Euclidean inner product).
Let denote the set of all real matrices that are strictly positive. I claim that is a convex open subset of . Fix , and suppose satisfies . Then
I claim that this last expression is strictly greater than zero for sufficiently small . Indeed, the map defines a continuous function on the unit sphere, which is compact in finite-dimensional spaces. Hence, the extreme value theorem implies that the infimum is attained at some , and the claim follows from our hypothesis that is strictly positive. Choosing , we obtain the desired conclusion. Hence, .
To see that is convex, fix and . Then for all ,
since both and are positive.
We now show that the function
Proof. Since any positive matrix is unitarily similar to a diagonal matrix, we may assume without loss of generality that is diagonal. Since is strictly positive, has an invertible self-adjoint square root, which we denote by . Hence,
If we make the change of variable , then
Now let . By Hölder’s inequality,
Using our computation above, we obtain the inequality
which gives, upon rearranging, the inequality
Taking the logarithm of both sides completes the proof of the concavity of .