The question I have been lost in for a while is when will a matrix have either all real or complex eigenvalues? (Depending on dimensions of the matrix in question, complex and real eigenvalues may coalesce, but the question remains the same.) I'm wondering how to recognize them and wish to have a list of different "types" of matrices that always yield either real of complex eigenvalues. Thanks!
2026-04-02 16:51:12.1775148672
Real or imaginary eigenvalues?
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An eigenvalue $\lambda$ of a real antisymmetric matrix will necessarily satisfy $\lambda+\bar\lambda=0$ and therefore is purely imaginary. As Mustafa pointed out in a comment, a real symmetric matrix will always have real eigenvalues.