HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 334

Singular value decomposition To understand the operation of Function SNRM, we need to introduce the concept of matrix decomposition. Basically, matrix decomposition involves the determination of two or more matrices that, when multiplied in a certain order (and, perhaps, with some matrix inversion or transposition thrown in), produce the original matrix. The Singular Value Decomposition (SVD) is such that a rectangular matrix Am×n is written as Am×n = Um×m ⋅Sm×n ⋅V Tn×n, Where U and V are orthogonal matrices, and S is a diagonal matrix. The diagonal elements of S are called the singular values of A and are usually ordered so that si ≥ si+1, for i = 1, 2, ..., n-1. The columns [uj] of U and [vj] of V are the corresponding singular vectors. (Orthogonal matrices are such that U⋅ UT = I. A diagonal matrix has non-zero elements only along its main diagonal). The rank of a matrix can be determined from its SVD by counting the number of non-singular values. Examples of SVD will be presented in a subsequent section. Functions RNRM and CNRM Function RNRM returns the Row NoRM of a matrix, while function CNRM returns the Column NoRM of a matrix. Examples, Row norm and column norm of a matrix The row norm of a matrix is calculated by taking the sums of the absolute values of all elements in each row, and then, selecting the maximum of these sums. The column norm of a matrix is calculated by taking the sums of the absolute values of all elements in each column, and then, selecting the maximum of these sums. Page 11-8