Low-rank approximation and Eckart-Young-Mirsky theorem for complex matrices

Question

For the classic problem of low-rank approximation:

\begin{equation} \begin{aligned} \min_{r(F) \leq k} \lVert X - F \rVert_F \end{aligned} \end{equation}

for a given constant real matrix $X$ approximated by $ F $ of rank at most $ k $. I know the Eckart-Young-Mirsky theorem states the solution when $ X $ is real but what about when $ X $ is a matrix with complex entries? Is the solution using SVD still the same as the Eckart-Young-Mirsky theorem? I am referring here to the Frobenius matrix norm which is well-defined for complex matrices as well and always positive. I wonder if Eckart-Young-Mirsky carries over to complex numbers for the Frobenius norm. I thank all helpers for any references to solutions for the complex problem.

Answer 1

The answer is yes: the result holds (with the same formula) if we discuss complex matrices under the Frobenius norm. As for a reference, note that the paper of Mirsky (the third name in the theorem) is about complex matrices. The citation:

The Quarterly Journal of Mathematics, Volume 11, Issue 1, 1960, Pages 50–59, https://doi.org/10.1093/qmath/11.1.50