easy to calculate: The off-diagonal elements of are zero; that is, is also a diagonal matrix. Therefore, calculating reduces to just calculating the function on each of the eigenvalues. A similar technique works more generally with the holomorphic functional calculus, using from above. Once again, we find that Examples which are examples for the functions . Furthermore, is the matrix exponential. Decomposition for special matrices When is normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Normal matrices A complex-valued square matrix is normal (meaning , where is the conjugate transpose) if and only if it can be decomposed as where is a unitary matrix (meaning ) and is a diagonal matrix. The columns u1, ..., un of form an orthonormal basis and are eigenvectors of with corresponding eigenvalues λ1, ..., λn. If is restricted to be a Hermitian matrix (), then has only real valued entries. If is restricted to a unitary matrix, then takes all its values on the complex unit circle, that is, . Real symmetric matrices As a special case, for every real symmetric matrix, the eigenvalues are real and the eigenvectors can be chosen real and orthonormal. Thus a real symmetric matrix can be decomposed as where is an orthogonal matrix whose columns are the real, orthonormal eigenvectors of , and is a diagonal matrix whose entries are the eigenvalues of . Useful facts Useful facts regarding eigenvalues The product of the eigenvalues is equal to the determinant of Note that each eigenvalue is raised to the power , the algebraic multiplicity. The sum of the eigenvalues is equal to the trace of Note that each eigenvalue is multiplied by , the algebraic multiplicity. If the eigenvalues of are , and is invertible, then the eigenvalues of are simply . If the eigenvalues of are , then the eigenvalues of are simply , for any holomorphic function . Useful facts regarding eigenvectors If is Hermitian and full-rank, the basis of eigenvectors may be chosen to be mutually orthogonal. The eigenvalues are real. The eigenvectors of are the same as the eigenvectors of . Eigenvectors are only defined up to a multiplicative constant. That is, if then is also an eigenvector for any scalar . In particular, and (for any θ) are also eigenvectors. In the case of degenerate eigenvalues (an eigenvalue having more than one eigenvector), the eigenvectors