eigenvalues of the sum of a diagonal matrix and a skew-symmetric matrix Suppose $A$ is a  skew-symmetric matrix (i.e., $A+A^{\top}=0$) and $D$ is a diagonal matrix. Under what conditions, $A+D$ is a Hurwitz stable matrix? 

I would like to give an answer that relies on the Lyapunov equation. 
The matrix $A+D$ satisfies the following Lyapunov equation:
$$(A+D)I+I(A+D)^T=2D$$
where $I$ is the identity matrix. Suppose $\lambda\in\sigma(A+D)$ and $x^*(A+D)=\lambda x^*$. Then
$$Re(\lambda)=\frac{1}{2}\frac{x^*(2D)x}{x^*x}$$


*

*If $D$ is negative definite, which means all the diagonal entries are negative, then $A+D$ is stable.

*If $D$ is negative semi-definite, which means all the diagonal entries are nonpositive, then $A+D$ is stable if and only if $(A+D, D)$ is controllable.

*If $D$ is not definite, which means the diagonal entries can be either positive, zero, or negative, then according to the above equation the matrix $A+D$ may not be stable.


Reference: book Topics in Matrix Analysis
