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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.

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  • $\begingroup$ Is $P$ itself symmetric? $\endgroup$ – Fimpellizieri Apr 12 '16 at 21:40
  • $\begingroup$ Yes, $P$ is also symmetric $\endgroup$ – Joy Yin Apr 12 '16 at 21:42
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A few things to consider: By definition of the inner product we have the rule that $\langle Ax, x\rangle = \langle x, A^Tx\rangle$ for all vectors $x$ and matrices $A$. Therefore we have $\langle C^TPCv, v\rangle = \langle PCv, Cv\rangle$ for all $v$. Let $y = Cv$ then your equation reduces to $\langle C^TPCv, v\rangle = \langle Py, y\rangle$.

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