# term for a sum of diagonal and skew-symmetric matrix?

Is there a term for a matrix that is a sum of a diagonal and a skew-symmetric matrix?

One particular example of this is a 2x2 matrix of the form

$$M = \begin{bmatrix} a & b \\ -b & a \end{bmatrix}$$

where a and b are real coefficients.

There's something special about this form vs. general matrices with off-diagonal coefficients, in the context of multivariable differential equations, that I can't put my finger on (something to do with the symmetry of the eigenvalues and eigenvectors), but maybe I can figure it out if I know what search term to use.

edit: magic property encountered here in the 2x2 case: the eigenvalues are $\lambda = a\pm jb$ and the eigenvectors are $v=\begin{bmatrix}1\\\pm j\end{bmatrix}$.

(p.s. yes, I'm an engineer, I use $j=\sqrt{-1}$ instead of $i$)

• Well, assuming the diagonal entries are all the same, the eigenvalues are purely imaginary numbers plus the diagonal entry of the diagonal matrix you add. – Nishant May 23 '14 at 23:32
• I gave an answer about the spectrum of the matrix here: math.stackexchange.com/questions/952233/… – Shiyu Dec 1 '15 at 23:59