# SVD of a specific matrix and Singular values behaviour

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.