I was trying to derive the upper bound for an equation. That equation consists these terms where I got stuck. My specific problem is here to find outs the upper bound for the given two mathematical terms $$- E[xy^T]A^T -AE[yx^T], $$ where $x$ and $y$ are vectors with zero mean, $A$ is any constant matrix, $E[xy^T] = P{xy}$ and $E[yx^T] = (Pxy)^T$.
I found one interesting formula in [1] which can be used to solve the above problem. For any two vectors $a, b \in \mathbb{R}^n$ $$ab^T + b a^T \leq \alpha aa^T + \alpha^{-1} bb^T, $$ where $\alpha (>0)$ is a positive constant.
After applying the above formula, the inequality becomes \begin{equation*} \begin{split} &-E[xy^T] A^T - A E[yx^T] \\& \leq \alpha E[xx^T] + \alpha^{-1} A E[yy^T]A^T \\& \leq \alpha P_{xx} + \alpha^{-1} A P_{yy} A^T. \end{split} \end{equation*}
But I want to get the above equation in cross-covariance form $(P_{xy})$.
Any suggestions are highly appreciated!
[1] L. Li, D. Yu, H. Yang, and C. Yan, "UKF for Nonlinear Systems with Event-triggered Data Transmission and Packet Dropout," in 3rd International Conference on Informative and Cybernetics for Computational Social, 2016.