MHB Proving Positive Definite Scalar Product for $n \times n$ Matrices

rputra
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Consider $X, Y$ as $n \times n$ matrices, I am given this definition of scalar product:
$$\langle X, Y \rangle = tr(X Y^T),$$
and I need to prove that it is positive definite scalar product. Of several properties I need to prove, two of them are
(1) $\langle X, X\rangle \geq 0$ and
(2) $\langle X, X\rangle > 0$ if $X \neq 0.$
I am lost on proving these two properties, any help or hints to prove them would be very much appreciated. Thank you before hand for your time and effort.
 
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Hi Tarrant,

A good place to start would be with the definition of the trace for an $n\times n$ matrix. Do you recall what this was?
 
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