# Factorization of a complex-valued positive matrix into product of Hankel matrices.

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.