I have a rectangular matrix $A \in \mathcal{M}_{l,n} (\mathbb{C})$, $l>n$ which has this property :

$$A=\left[\begin{matrix} M_1 & i M_2 \\ M_2 & -i M_1 \end{matrix}\right]$$

Where $M_i$ are (complex) rectangular matrices with same size $\frac l 2 \times \frac n 2$.

Can i expect (formally) some special behaviour from the singular values of $A$ relatively to its size ?

Some Background :

Numerically, I noticed that the smallest singular values of $A$ are very close to $0$ relatively to the greatest ones when $n < l < 2n$ and not when $l \ge 2n$. Then I found out that $A$ has that special structure described at the beginning of that question. I do expect a (quantitative) link between those two things.

My handwavy explanation is that, when $n < l < 2n$, the redundacy of information encoded by the special structure is well captured by the SVD, so we do not need all the eigenvalues to construct a good approximation of $A$.

Thank you.


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