# Maximum eigenvalue of a skew symmetric matrix

There is a nice method to find the maximum eigenvalue of a real symmetric matrix:

Let $$A$$ be a real symmetric $$n\times n$$ matrix. Then the maximum eigenvalue of $$A$$ is given by, $$\lambda_{\max}=\max_{\Vert x\Vert=1}x^TA\,x.$$

This is of course, quite easy to prove using the spectral theorem.

I was wondering if there is a similar result for the maximum absolute value of the eigenvalue of a real skew symmetric matrix?

So you are looking for spectral radius of skew symmetric matrix.

$$||A||_2 = \sqrt{\rho(A^TA)} = \sqrt{\rho(-A^2)} = \sqrt{\rho(A^2)} = \rho(A)$$.

That means maximum absolute eigenvalue of skew symmetric matrix is just L2 norm of it.

• How did you write the first equation, $\Vert A\Vert_2=\sqrt{\rho(A^TA)}$?
– R_D
May 2, 2021 at 18:16
• It is a well known equation. You can find proof here at page 5 (proof is simple): math.drexel.edu/~foucart/TeachingFiles/F12/M504Lect6.pdf May 2, 2021 at 18:20
• Since your matrix is real, Hermitian conjugate ($A^*$) of matrix is just transpose ($A^T$) of matrix. May 2, 2021 at 18:21

1.) change the field to $$\mathbb C$$ and $$B:=i \cdot A$$ is Hermitian.
2.) Now reuse your max formula for Hermitian matrices and apply it to $$B$$.
3.) recall that scaling a matrix by any number on the unit circle doesn't change the modulus of any eigenvalues.