# Expectation of product of hermitian matrix

Let $\textbf{h} \in \mathbb{C}^{N\times 1}$, $\textbf{a} \in \mathbb{C}^{N\times 1}$, $\textbf{b} \in \mathbb{C}^{N\times 1}$ and $\textbf{c} \in \mathbb{C}^{N\times 1}$. The variable $h_i$ is defined as $h_i=\left(a_i+b_i\right)c_i$

$\textbf{a}=[1~2~3~4]^T$, $\textbf{b}=[5~6~7~8]^T$, $\textbf{c}=[9~10~11~12]^T$, Then $h_1$ is calculated as follows $(a_1+b_1)c_1=(1+5)\times 9$, $h_2 =(a_2+b_2)c_2=(2+6)\times 10$, $h_3 =(a_3+b_3)c_3=(3+7)\times 11$, $h_4 =(a_4+b_4)c_4=(4+8)\times 12$.

Finally $\textbf{h}=[h_1, h_2, h_3, h_4]^T$.

Question: 1. $\textrm{E}\left[\textbf{h}\textbf{h}^H\right]$=? in terms of $\textrm{E}\left[\textbf{c}\textbf{c}^H\right]$

2.Can I write $\textbf{h}$ in terms of $\textbf{a}$ $\textbf{b}$, $\textbf{c}$?

You could define a diagonal matrix $\mathbf{D}$ with diagonal elements $d_{ii}=a_i+b_i$. Then vector $\mathbf{h}$ can be written as
$$\mathbf{h}=\mathbf{D}\mathbf{c}$$
$$\mathbb{E}[\mathbf{h}\mathbf{h}^H]=\mathbf{D}\mathbb{E}[\mathbf{c}\mathbf{c}^H]\mathbf{D}^H$$
(I assume that $\mathbf{a}$ and $\mathbf{b}$ are deterministic and $\mathbf{c}$ is random, because you said you wanted the expectation in terms of $\mathbf{c}$.)