Do you have an example of a real skew-symmetric matrix (seen as an operator over $\mathbb{C}^n$) having at least one (purely imaginary) eigenvalue with algebraic multiplicity strictly greater than the geometric one?
Skew-symmetric non-diagonalizable matrix
diagonalizationlinear algebramatricessymmetric matrices
Best Answer
Such a matrix doesn't exist. Since it is skew-symmetric, it's normal and therefore diagonalizable.