 If instead is further restricted to a unitary matrix (), takes all its values on the complex unit circle, that is, .As a special case, for every real symmetric matrix, the eigenvalues are real and the eigenvectors can be chosen such that they are orthogonal to each other.Thus a real symmetric matrix can be decomposed as

where is an orthogonal matrix whose columns are the eigenvectors of , and is a diagonal matrix whose entries are the eigenvalues of .