5. Real Rectangular Matrices
Society for Industrial and Applied Mathematics eBooks · 2002
SECTION 5.1 INTRODUCTION If A is a ℓxn matrix then the singular value decomposition of A has the following form. A=YΣ XH 5.1.1 where Y is a ℓxℓ unitary matrix, X is a nxn unitary matrix and for ℓ≥n, Σ is a real rectangular diagonal matrix of the form Σ= [ Σ1 0] . 5.1.2 Σ1 is an nxn diagonal matrix with nonnegative entries and 0 denotes the (ℓ−n)xn matrix of zeros. For ℓ≤n, Σ would be of the form [Σ1 0]. To simplify the discussion in this Chapter we will assume that ℓ≥n. This is typical in the applications. For example in fitting data, we would typically have more measurements than fitting parameters. The corresponding statements for the case ℓ≤n can be obtained directly from the arguments given, if these arguments are applied to AH rather than to A. If A is real then Y and X are orthogonal matrices. We have that AHA = XΣTΣXH and AAH = YΣΣTYH. Two equivalent forms of Eqn(5.1.1) which are easily obtained from that equation are AX=YΣand AH Y=X ΣT . 5.1.3 Thus, the Y span the range or column space of A and the X span the range of AH. In this chapter we present a single-vector Lanczos procedure with no reorthogonalization for computing singular values and corresponding singular vectors of real rectangular matrices. First in Section 5.2 we summarize some of the properties of singular values and briefly discuss their relationship to the eigenvalues of the given matrix. Second in Section 5.3 we briefly discuss several applications. Finally in Section 5.4 we describe our Lanczos procedure. FORTRAN code for this procedure is provided in Chapter 6 of Volume 2 of this book. Theorem 5.1.1 states that the matrix decomposition given in Eqn(5.1.1) exists for any matrix. Proofs of this Theorem can be found in many places, see for example Stewart [1973, p. 319] or Lawson and Hanson [1974, p. 19]. THEOREM 5.1.1 Let A be a ℓxn matrix, then there are unitary matrices Y and X and a rectangular ℓxn diagonal matrix Σ satisfying Eqn(5.1.1).