Hello Shu-Chang,

Every diagonalizable matrix A can be decomposed to A=Q L Q^{-1} where L is

diagonal eigenvalue matrix, Q is eigenvector matrix.

Otherwise:

Every 2n × 2n real skew-symmetric matrix can be written in the form A = Q Σ Q^T where Q is orthogonal. See for details:

Youla, D. C. (1961). "A normal form for a matrix under the unitary congruence group". Canad. J. Math. 13: 694–704.

Hi Gro,

Thank you for your answer. 2n × 2n real skew-symmetric matrix could be decomposed to something like symplectic matrix, what would it be if order of matrix is 2n+1?

In the odd-dimensional case the same decomposition is possible, but Σ has at least one row and column of zeros.