Why is this matrix symmetric?

Question

There is an example in Boyd's convex optimization lecture notes. He just said in the lecture that the matrix which is underlined in red color is symmetric! How can we claim that when there is no assumption on the $V$?

---

Answer 1

Read the last line:

(for any choice of $X\succ0,V$)

So, somewhere in the lecture notes the author must have assumed that $X$ is positive definite and $V$ is symmetric. Now, when $X$ is positive definite, the usual convention is that $X^{1/2}$ denotes not an arbitrary square root of $X$ but the unique positive definite square root of $X$. Hence $X^{-1/2}$ and in turn $X^{-1/2}VX^{-1/2}$ are symmetric.