How to Write a Vector Multiplication as a Trace of Matrix? Let $\mathbf{w}_j\in\mathbb{C}^{M\times 1}$ and $\mathbf{h}_k\in\mathbb{C}^{M\times 1}$ be two complex vectors.
How to prove this?
$$\mid\mathbf{w}_j^\dagger\cdot\mathbf{h}_k\mid^2=\mathrm{Tr}\left(\mathbf{w}_j\cdot\mathbf{w}_j^\dagger\cdot\mathbf{h}_k\cdot\mathbf{h}_k^\dagger\right)$$
For $M=2$, I calculate:
$$\begin{align}
   \begin{bmatrix}
           w_{j1} \\
           w_{j2} \\
         \end{bmatrix}
  \cdot[\bar{w}_{j1}, \bar{w}_{j2}]\cdot
   \begin{bmatrix}
           h_{k1} \\
           h_{k2} \\
         \end{bmatrix}
  \cdot[\bar{h}_{k1}, \bar{h}_{k2}]\end{align}$$
And I found $$\mathrm{Tr}\left(\mathbf{w}_j\cdot\mathbf{w}_j^\dagger\cdot\mathbf{h}_k\cdot\mathbf{h}_k^\dagger\right)=|w_{j1}|^2\cdot |h_{k1}|^2+|w_{j2}|^2\cdot |h_{k2}|^2+\overline{w_{j1}\cdot h_{k2}}\cdot w_{j2}\cdot h_{k1}+w_{j1}\cdot h_{k2}\cdot \overline{w_{j2}\cdot h_{k1}}$$

I think I have found the answer.
We have, for $\mathbf{a}\in\mathbb{C}^{M\times1}$:
$$\mathrm{Tr}\left(\mathbf{a}\cdot \mathbf{a}^\dagger\right)=|\mathbf{a}|^2$$
Let $\mathbf{a}=\mathbf{w}_j^\dagger\cdot\mathbf{h}_k$, then:
$$\mathrm{Tr}\left(\mathbf{w}_j^\dagger\cdot\mathbf{h}_k\cdot \left(\mathbf{w}_j^\dagger\cdot\mathbf{h}_k\right)^\dagger\right)=\mathrm{Tr}\left(\mathbf{w}_j^\dagger\cdot\mathbf{h}_k\cdot \mathbf{h}_k^\dagger\cdot\mathbf{w}_j\right)$$
Applying the fact that $\mathrm{Tr}(A\cdot B)=\mathrm{Tr}(B\cdot A)$, as suggested, we find the result.
