I have a vector containing all the PSD of a series of signals

$\underline{s} = \left[ |S_{x_1x_1}| |S_{x_2x_2}| |S_{x_3x_3}| ...\right]^T$

I also have a matrix $\gamma^2$ containing all the coherences

$\gamma_{ij} = \frac{|S_{x_ix_j}|^2}{S_{x_ix_i} S_{x_jx_j}}$

I would like to obtain the matrix of the cross-spectral densities, i.e. a matrix $\mathbf{S}$ so that $S_{ij} = |S_{x_ix_j}|^2$

Is it possibile to obtain it from $\underline{s}$ and $\mathbf{\gamma}$ using matrix notation?

The only way I found at the moment is:

$\mathbf{S} = (s s^T) \circ \gamma$

where $\circ$ indicates the Hadamard (element-wise) power and multiplication. This is however seldom used and usually element-wise multiplication is avoided in linear algebra.

Is there another way to do it?


$S= diag(\underline{s}) \cdot \underline{\underline{\gamma}} \cdot diag(\underline{s})$


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