which matrices has all their eigenvalues as real numbers?

Answer 1

an example is symmetric matrices

Answer 2

Every eigenvalue of a self-adjoint operator (matrix) is real.

Self adjoint means T=T*, ie. the matrix is equal to it's adjoint.

IIf, and only if, the transformation is with respect to an orthonormal basis, then the adjoint is the complex conjugate transpose.

The previous poster is correct for the real case. Assuming the matrix is real and is w.r.t. an orthonormal basis, then it's equal to its complex conjugate transpose, ie, symmetrical and thus has real eigenvalues.