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I am dealing with a sort of "variance-covariance" matrix that surges from the multiplication of the Hankel matrix of a complex-valued time series and it's conjugate transpose. $$R_X=\mathcal{H}\mathcal{H}^*$$ (where * represents the conjugate transpose).


This part is not important for the question but may offer some context.

I estimate an eigendecomposition of the $R_X$ matrix such that I end up with

$$R_X=U\cdot (\Lambda_X +\sigma^2\cdot I)\cdot U^T$$

where $\Lambda_X$ and $\sigma^2\cdot I$ are the eigenvalues of the variance-covariance matrices of two independent sources.

By some means I manage to estimate $\sigma^2\cdot I$ such that I could obtain a new matrix $R_{X2}=U\cdot (\Lambda_X)\cdot U^T$.


Theoretically $R_{X2}$ is also the result of multiplying the Hankel matrix of a complex-valued time series and it's conjugate transpose.

$$R_{X2}=\mathcal{H_2}\mathcal{H_2}^*$$

But I don't know $\mathcal{H_2}$.

So this lead me to the original question: Is there any way of obtaining $\mathcal{H_2}$ knowing $R_{X2}$? Where the latter is a complex-valued positive semidefinite matrix?

Such a decomposition of a matrix into the product of a Hankel matrix and it's conjugate transpose is unique?

Any hint or suggestion will be greatly appreciated.

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