Block Semidefnitive Matrix

Question

Suppose we have a semidefinite matrix named $Q$. If I build a matrix out of it like this:

$$M = \pmatrix{Q&-Q\\-Q&Q}$$

Is $M$ semidefinite too? I have faced this problem working on support vector regression just in case you may want to know.

Answer 1

If we have a vector $v=\pmatrix{a\\b}$, then $$v^tMv=a^tQa-a^tQb-b^tQa+b^tQb=(a^t-b^t)Q(a-b)=(a-b)^tQ(a-b)\ge0.$$