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 '14 at 23:32
  • $\begingroup$ I gave an answer about the spectrum of the matrix here: math.stackexchange.com/questions/952233/… $\endgroup$ – Shiyu Dec 1 '15 at 23:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.