Since a real matrix can have complex eigenvalues (occurring in complex conjugate pairs), even for a real matrix A, U and T in the above theorem can be complex.

Can real eigenvalues have complex eigenvectors?

If the n × n matrix A has real entries, its complex eigenvalues will always occur in complex conjugate pairs. ... This is very easy to see; recall that if an eigenvalue is complex, its eigenvectors will in general be vectors with complex entries (that is, vectors in Cn, not Rn).

Can a matrix have no real eigenvalues?

There is at Least One Real Eigenvalue of an Odd Real Matrix Let n be an odd integer and let A be an n×n real matrix. Prove that the matrix A has at least one real eigenvalue.