TABLE OF CONTENTS A matrix A is diagonalizable if and only if A possesses a complete independent set of eigenvectors, and if such a complete set is used for columns of P , then P ?1 AP = D is diagonal (p. 507). But even when A possesses a complete independent set of eigenvectors, there's no guarantee that a complete orthonormal set of eigenvectors can be found. In other words, there's no assurance that P can be taken to be unitary (or orthogonal). And the Gram-Schmidt procedure (p. 309) doesn't help-Gram-Schmidt can turn a basis of eigenvectors into an orthonormal basis but not into an orthonormal basis of eigenvectors. So when (or how) are complete orthonormal sets of eigenvectors produced? In other words, when is A unitarily similar to a diagonal matrix?
A ? C n n is unitarily similar to a diagonal matrix (i.e., A has a complete orthonormal set of eigenvectors) if and only if A* A = AA*, in which case A is said to be a normal matrix.
Whenever U* AU = D with U unitary and D diagonal, the columns of U must be a complete orthonormal set of eigenvectors for A, and the diagonal entries of D are the associated eigenvalues.
Proof. If A is normal with ? ( A) = {
