what can we say about $\mathrm{E}(y^HC_{yy}^{-1}y)$

Question

suppose $y$ is a continuous random vector with zero mean, but with unkonwn(or arbitrary) pdf, $C_{yy}$ is the covariance matrix of y, then what can we say about $$ \mathrm{E}(y^HC_{yy}^{-1}y) $$ ? $E(.)$ is the expected value.

Thank you!

Answer 1

Note that we have \begin{align*} \def\E{\mathbf E}\E[y^* C_{yy}^{-1}y] &= \E\left[\sum_{i,j} \bar y_i \bigl(C_{yy}^{-1}\bigr)_{ij} y_j\right]\\ &= \sum_{i,j} \bigl(C_{yy}^{-1}\bigr)_{ij} \E[\bar y_i y_j]\\ &= \sum_{i,j} \bigl(C_{yy}^{-1}\bigr)_{ij} \bigl(C_{yy}\bigr)_{ji}\\ &= \operatorname{tr} C_{yy}^{-1}C_{yy}\\ &= n. \end{align*}