I got stuck in the following problem, any help would surely be appreciated:
I have to prove that a block matrix of the form:
is positive definite, where is symmetric and positive definite, and is anti-symmetric, that is, .
When I take the following nonzero vector:
and try to check the usual condition , I get the following:
And is a scalar, so:
Substituting above, we have:
Now, how can I prove that the real number is always positive?
For sure, and are both positive, or at least one of them is positive and the other is zero, because is positive definite.