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A Hermitian Matrix is defined as a self-adjoint matrix (so the complex conjugate transpose of M is equal to M). But the usefulness of Hermitian Matrices seems to be that they necessarily have orthogonal eigenvectors that span the space and that all eigenvalues are real.

If I tell you I have a matrix whose eigenvectors are orthogonal, span the space, and whose eigenvalues are all real, does this necessarily mean I have a Hermitian matrix? If not, is there a general name for such a matrix?

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If $M = UDU^*$ where $U$ is unitary and $D$ is a real diagonal matrix, we have $$M^* = (UDU^*)^* = (U^*)^*D^*U^* = UDU^* = M,$$ so $M$ is self-adjoint.

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