Do matrices that have block structure made of identity matrices with negative identity matrices along the main diagonal have a name? More specifically, do matrices such as:
$$ {{A}_{n}}=\left[ \begin{matrix} -{{I}_{n}} & {{I}_{n}} \\ {{I}_{n}} & -{{I}_{n}} \\ \end{matrix} \right] $$
have a particular name? Here $n$ is a nonzero positive integer and $I_n$ is an identity matrix of dimension $n$.
More generally do matrices that are "quasi-similar" to such these matrices have a name? $A_n$'s have $n$ zero eigenvalues and the remaining $n$ just equal to $-2$. Matrices actually similar to $A_n$ then should have the same eigenvalues. If we flip the signs of the blocks on each row, the non-zero but same-valued eigenvalues change signs.
By my "quasi-similar" I mean the same eigenvalue structure: $n$ zero eigenvalues and $n$ nonzero eigenvalues (a. possibly same, [but all negative or all positive] and b. possibly different [$n-1$ repeats of the non-zero ones are allowed as along as all $n$ have the same sign]).
I suspect this is related to symplectic matrices, but I am not familiar with the theory. The background of the question is that it arises from center-manifold theory and in particular systems that have only stable modes and a slow manifold.
References to related topics/matrices are also welcome in the comments.
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EDIT: Maybe I should clarify that I am not looking for a "name"/name. In particular; if there is no name, then the accepted answer should be "There is no name for such kind of matrices". But that doesn't help me or future investigators: Instead I would rather prefer (if in fact there is no name) an answer that elucidates relationships with other groups/matrices/structures that do have a name.