When EIG is called with an exactly symmetric/hermitian matrix, MATLAB falls back to a specialized algorithm that guarantees that U is orthogonal/unitary and that D is real. There is no such special algorithm choice for skew-symmetric matrices, so there is no guarantee here, even though if the problem is nicely conditioned, the result will be close to that:
>> rng default; A = randn(10); A = A - A';
>> [U, D] = eig(A);
>> max(abs(real(diag(D))))
ans =
2.1034e-16
>> norm(U'*U - eye(10))
ans =
4.7239e-15
However, if matrix B is (exactly) skew-symmetric, it implies that matrix A = 1i*B is hermitian, and passing this matrix to EIG will result in unitary eigenvectors and all-real eigenvalues, which you can then transform back:
[U, D] = eig(1i*A);
D = -1i*D;
>> max(abs(real(diag(D))))
ans =
0
>> norm(U'*U - eye(10))
ans =
1.5001e-15
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