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I've encountered the matrix below and I'm curious about its properties;

[tex] R=

\begin{pmatrix}

0 & N-S\\

N+S & 0

\end{pmatrix}

[/tex]

where R, N and S are real matrices, R is 2n by 2n, N is n by n symmetric and S is n by n skew-symmetric.

Clearly R is symmetric so the eigenvalues are real, but what else can be said about a matrix of this type? I checked through some literature but didn't really know what to look for. Surely the form is simple enough that it should have been studied.

In a special case, the elements of the rows of the matrix N+S sum to zero. Could this affect the properties somehow?

Any ideas would be much appreciated!

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# Properties of a special block matrix

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