A problem related to semi definite matrix Let $A\in \mathbb R^{n\times d}$, and $Y,Z$ are two matrices in $\mathbb R^{{d\times j}}$, then there is a matrix $P$ such that $P=A^TA(YY^T-ZZ^T)$. Then my question is what properties should matrix $Y$ and $Z$ satisfy so that matrix $P$ is positive semi definite.
 A: We know these facts: 


*

*$X^TX$ is PSD for any $X$

*The product of two symmetric PSD matrices is PSD, iff the product is also symmetric. This equivalence is not in Horn and Johnson so here's the reference.
Reference: On a product of positive semidefinite matrices, A.R. Meenakshi, C. Rajian, Linear Algebra and its Applications, Volume 295, Issues 1–3, 1 July 1999, Pages 3–6.

*The difference of PSD matrices is not guaranteed to be PSD. Weyl's inequalities are a simple way to create bounds. For example if the maximum eigenvalue of $ZZ^T$ is lesser than the minimum eigenvalue of $YY^T$ then $YY^T - ZZ^T$ is guaranteed to be PSD.


Combining these facts we can say that
$A^TA(YY^T - ZZ^T)$ is PSD if both $A^TAYY^T$ and $A^TAZZ^T$ are symmetric (also fine if $A^TA$, $YY^T$ and $ZZ^T$ are simultaneously diagonalizable or if $A^TA$ commutes with $YY^T$ or $ZZ^T$), and the eigenvalues of $A^TAYY^T$ are suitably larger than the eigenvalues of $A^TAZZ^T$.
or if the eigenvalues of $YY^T$ are larger than eigenvalues of $ZZ^T$ and product of $A^TA$ and $(YY^T - ZZ^T)$ is symmetric. 
