Spectral theorem

The finite-dimensional real version of the spectral theorem is a theorem of linear algebra that says that any real symmetric matrix can be diagonalized by an orthogonal matrix. In other words, if M is a matrix whose entries are real numbers, and M^T=M (i.e., M is symmetric) then there is some matrix G with real entries such that G^TG=GG^T=I (i.e., G is orthogonal) and there is some diagonal matrix D, such that

• G^TMG=D, or, equivalently
• M=GDG^T.

The entries in the diagonal matrix D are the eigenvalues of M. In effect, this means that the eigenvalues are real and the eigenvectors are mutually orthogonal. The finite-dimensional complex version of the spectral theorem says that any Hermitian matrix can be diagonalized by a unitary matrix, again implying that the eigenvalues are real and the eigenvectors are orthogonal.

An infinite-dimensional counterpart is a theorem of functional analysis.