Matrix Analysis

Introduction

In Chapter 1, we made an initial study of similarity of A ∈ Mn via a general nonsingular matrix S, that is, the transformation A → S−1AS. 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 ∈ Mn 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 ∈ Mn, m, the transformation A →U AV, in which U ∈ Mm and V ∈ Mnare 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 ∈ Mn, m is unitarily equivalent to a nonnegative diagonal matrix whose diagonal entries (the singular values of A) are of great importance.