- #1
rputra
- 35
- 0
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.
$$\langle X, Y \rangle = tr(X Y^T),$$
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.