Let $Q$, of dimension $m \times n$, be a matrix with i.i.d. complex Gaussian elements; we suppose that these elements have zero mean and have the same variance $= v$.
We define $A$ as an $n \times k$ matrix, for which each column $\mathbf{a}_i$ (of dimension $n \times 1$), for $i=1,\ldots,k$, we have $\mathbf{a}^h_i\mathbf{a}_i=|\mathbf{a}_i|^2=1$; the superscript $h$ is used to denote the conjugate transpose.
My aim is to find the expectation of $Q A (Q A)^h$: $E[Q A A^h Q^h]$. Any idea ?
Note that we assume $Q$ and $A$ are independent.
I am also interested in the result if the elements of $Q$ have a mean $\mu$ ($\ne 0$).
We know: $E[Q \mathbf{u} \mathbf{u}^h Q^h]=v \mathbf{I}_m$ if $\mathbf{u}$ is a unit norm vector.