Eigenvalues of the sum of a symmetric and a anti-symmetric matrix

Question

Let $M_S$ be a symmetric and negative definite matrix and $M_A$ be an anti-symmetric matrix, both in $\mathbb{R}^{n\times n}$. The eigenvalues of $M_S$ lie in the negative part of the complex plane, while the eigenvalues of $M_A$ are found on the imaginary axis.

I'm interested in bounds for the eigenvalues of $M=M_S+M_A.$

Specifically, I want to know if we can say that the eigenvalues of $M$ will lie in the negative half of the complex plane. I thought this to be likely since $$\mathrm{tr}(M)= \mathrm{tr}(M_S)+\mathrm{tr}(M_A)=\mathrm{tr}(M_S),$$ i.e. it doesn't depend on $M_A,$ but I don't know how to prove it.

See: If $A+A^T$ is negative definite, then the eigenvalues of $A$ have negative real parts? — Ben Grossmann, Oct 01 '17 at 13:48
@Omnomnomnom Thanks for the link. I don't see the connection, however, since $M,$ $M_S$ and $M_A$ are completely unrelated, i.e., $M_S$ is not the symmetric part of $M,$ etc. Come to think of it, I know that $M=\frac{1}{2}(M+M^T)+\frac{1}{2}(M-M^T),$ but does that imply that $M_S=\frac{1}{2}(M+M^T)$ and $M_A=\frac{1}{2}(M-M^T)$? — Bobson Dugnutt, Oct 01 '17 at 19:52
Yes, we can deduce that $M_S = \frac 12 (M + M^T)$. Notably, the symmetric and skew-symmetric matrices form two subspaces whose direct sum is $\Bbb R^{n \times n}$. — Ben Grossmann, Oct 01 '17 at 20:02
Or, if you prefer: note that $$ (M_S + M_A) + (M_S + M_A)^T = M_S + M_A + M_S^T + M_A^T = M_S + M_A + M_S - M_A = 2M_S $$ — Ben Grossmann, Oct 01 '17 at 20:03
@Omnomnomnom Huh, well that gives the result directly, thanks once again! (Again, feel free to post as an answer if you please.) — Bobson Dugnutt, Oct 01 '17 at 20:05

score 1 · Accepted Answer · answered Oct 01 '17 at 20:08

1

To summarize the discussion in the comments:

It is true that the eigenvalues of $M$ must lie in the left half of the complex plane because the symmetric matrix $M + M^T = 2M_S$ is negative definite. The fact that $M + M^T$ is negative definite implies that $M$ has eigenvalues with negative real part is explained in this question/answer.

answered Oct 01 '17 at 20:08

Ben Grossmann

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I've asked another question here, that could be considered an extension of this one. I'd be very grateful if you'd take a look at it. Thanks. – Bobson Dugnutt Oct 02 '17 at 15:49

Eigenvalues of the sum of a symmetric and a anti-symmetric matrix

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