If you form the block matrix Y = (real(A), -imag(A) ; -imag(A), -real(A)), the real valued eigenvectors and eigenvalues of Y correspond to conjugate eigenvectors and eigenvalues of A; the complex conjugate pairs of eigenvalues of Y don’t correspond to anything. Then, for any angle theta, you can multiply a conjugate eigenvector v of A by exp(i theta) to get a new conjugate eigenvector, and find its associated eigenvalue by elementwise dividing A conj(exp itheta v) by v. The conjugate eigenvalues form up to 10 continuous rings around the origin,
If you form the block matrix Y = (real(A), -imag(A) ; -imag(A), -real(A)), the real valued eigenvectors and eigenvalues of Y correspond to conjugate eigenvectors and eigenvalues of A; the complex conjugate pairs of eigenvalues of Y don’t correspond to anything. Then, for any angle theta, you can multiply a conjugate eigenvector v of A by exp(i theta) to get a new conjugate eigenvector, and find its associated eigenvalue by elementwise dividing A conj(exp itheta v) by v. The conjugate eigenvalues form up to 10 continuous rings around the origin,