Could someone help prove the following: I have two independent random vectors $\mathbf{a} \in \mathbb{C}^{M \times 1}$ and $\mathbf{b}\in \mathbb{C}^{N \times 1}$. Both $\mathbf{a}$ and $\mathbf{b}$ have unit norm. There is a matrix $\mathbf{H} \in \mathbb{C}^{M \times N}$. Every element of $\mathbf{H}$ is an i.i.d. complex Gaussian random variable with zero mean and unit variance. Could someone show why $\mathbf{a}^{H}\mathbf{H}\mathbf{b}$ is a comlex gaussian variable with unit variance?
Thinks in advance!
I have an approach that seems to work. Write $a$ and $b$ as products of random unitary matrices $U_a \in U(M)$ $U_b \in U(N)$ with uniform (Haar) measure on the unitary group and $e_1=(1,0,0,\dots)$. $$ a = U_a e_1 $$ $$ b = U_b e_1 $$
Now the product you want is: $$ a^H Hb = e_1^TU_a^HH U_b e_1 $$ $H$ is a random unitary matrix with Gaussian measure $\propto \exp(-||H ||_F^2)$. This measure is invariant under left and right unitary transformations and therefore $U_a^HH U_b$ has the same distribution as $H$. As a result, your product is equivalent to any entry in $H$: a complex Gaussian w/ zero mean and unit variance.