Unitary Similarity and Unitary Equivalence

Abstract

Introduction In Chapter 1, we made an initial study of similarity of A ∈ M n via a general nonsingular matrix S , that is, the transformation A → S −1 AS . For certain very special nonsingular matrices, called unitary matrices , the inverse of S has a simple form: S −1 = S *. Similarity via a unitary matrix U , A → U * AU , is not only conceptually simpler than general similarity (the conjugate transpose is much easier to compute than the inverse), but it also has superior stability properties in numerical computations. A fundamental property of unitary similarity is that every A ∈ M n is unitarily similar to an upper triangular matrix whose diagonal entries are the eigenvalues of A . This triangular form can be further refined under general similarity; we study the latter in Chapter 3. The transformation A → S * AS , in which S is nonsingular but not necessarily unitary, is called * congruence ; we study it in Chapter 4. Notice that similarity by a unitary matrix is both a similarity and a *congruence. For A ∈ M n, m , the transformation A → U AV , in which U ∈ M m and V ∈ M n are both unitary, is called unitary equivalence . The upper triangular form achievable under unitary similarity can be greatly refined under unitary equivalence and generalized to rectangular matrices: Every A ∈ M n, m is unitarily equivalent to a nonnegative diagonal matrix whose diagonal entries (the singular values of A ) are of great importance.