Show that $$\langle\ \phi,X^TX+Y^TY - (Y^TX)^T - Y^TX \rangle\ \geq 0$$ where $\phi$ is a symmetric positive semidefinite matrix and $X, Y \in R^{m \times n}$.
Is my reasoning correct?
$$\langle\ \phi,X^TX+Y^TY - (Y^TX)^T - Y^TX \rangle\ =\operatorname{tr}(\phi X^T (X-Y) ) + \operatorname{tr}(\phi Y^T (Y-X) ) =\\ =\operatorname{tr}( \phi (X-Y)^T(X-Y))\geq 0$$
It's correct provided that the inner product is defined by $\langle A,B\rangle=\operatorname{tr}(AB^T)$ (or equivalently, by $\langle A,B\rangle=\operatorname{tr}(A^TB)$), but the proof is more complete if you explain why the last inequality holds: by the tracial property, $\operatorname{tr}(\phi(X-Y)^T(X-Y))=\operatorname{tr}((X-Y)\phi(X-Y)^T)$ and the latter trace is $\ge0$ because $(X-Y)\phi(X-Y)^T$ is positive semidefinite.