Chapter 24

Singular Values and Singular Value Inequalities

Roy Mathias

University of Birmingham

24.1 Definitions and Characterizations

Singular values and the singular value decomposition are defined in Section 5.6. Additional information on computation of the singular value decomposition can be found in Chapter 58. A brief history of the singular value decomposition and early references can be found in [HJ91, Chap. 3].

Throughout this chapter, q = min{m, n}, and if A ∈ ℂn×n has real eigenvalues, then they are ordered λ1(A) ≥ ⋯ ≥ λn(A).

Definitions:

For A ∈ ℂm × n, define the singular value vector sv(A) = (σ1(A),..., σq(A)).

For A ∈ ℂm × n, define r1(A) ≥ ⋯ ≥ rm (A) and c1 (A) ≥ ⋯ ≥ cn (A) to be the ordered Euclidean row and column lengths of ...

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