I have some difficulties to understand the concept of positive semi definite matrix, I read through the textbook and additional information that I found online, but still the concept is not really clear in my head. There is one practice problem which I am not sure how to approach.
Question: Let $P$ be a positive definite $n \times n$ matrix. If $C$ is any real $n\times m$ matrix, then show that $C^TPC$ is symmetric and positive semi definite.
The definition that we are given for positive semi-definite is "A self-adjoint operator $T$ on an inner product space $V$ is positive-semi definite if we have $\langle T(v),v\rangle \geq 0$".
Are all positive-semi-definite matrix are symmetric? Are they inherited all the properties and characterizations from symmetric matrix?
Any help will be really appreciated!
Thanks a lot.