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 $i$)

  • $\begingroup$ 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. $\endgroup$
    – Nishant
    May 23, 2014 at 23:32
  • $\begingroup$ I gave an answer about the spectrum of the matrix here: math.stackexchange.com/questions/952233/… $\endgroup$
    – Shiyu
    Dec 1, 2015 at 23:59


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