Singular value decomposition orthogonal matrices

In practice, the transformation matrix M can be found numerically from the singular value decomposition of X. The rank of X is equal to the number of non-zero singular values, and the transformation matrix corresponds to the transpose of die premultiplier orthogonal matrix. This process is illustrated below for die non-elementary reaction case. 

Using an orthogonal matrix U, we can rewrite the Singular Value Decomposition of a rectangular matrix A in the form ... 

Figure E-1 Singular value decomposition of a rectangular matrix A, using a column-orthogonal matrix U and a square diagonal matrix Q.

The analysis of a series of chiroptical spectra and recovery of systematic trends in a given set can be carried out in several ways. In the past, the results strongly depended on the spectroscopist s personal experience actually, this was the least objective part of the circular dichroism application. Nowadays, we can rely on general procedures of statistical data treatment like singular value decomposition, factor analysis (especially its first part, analysis of the correlation matrix and the projection of the experimental spectra onto the space of orthogonal components), cluster analysis and the use of neural networks. This field has been pioneered by Pancoska and Keiderling [72-76], and also by Johnson [77] when analyzing the chiroptical properties of biopolymers. 

Principal Components Analysis (PCA) is a multivariable statistical technique that can extract the strong correlations of a data set through a set of empirical orthogonal functions. Its historic origins may be traced back to the works of Beltrami in Italy (1873) and Jordan in Prance (1874) who independently formulated the singular value decomposition (SVD) of a square matrix. However, the first practical application of PCA may be attributed to Pearson s work in biology [226] following which it became a standard multivariate statistical technique [3, 121, 126, 128]. 

A little more expensive [n (m + I7nl3) flops and 2mn space versus n (m—nl3) and mn in the Householder transformation] but completely stable algorithm relies on computing the singular value decomposition (SVD) of A. Unlike Householder s transformation, that algorithm always computes the least-squares solution of the minimum 2-norm. The SVD of an m x n matrix A is the factorization A = ITEV, where U and V are two square orthogonal matrices (of sizes mxm and nxn, respectively), U U = Im, y V = In, and where the m x n matrix S... 

Hennessey and Johnson [81H1, 90J2] also directly analyze the CD spectrum of a protein from the CD spectra of reference proteins. By using singular value decomposition they create orthogonal CD basis vectors from the spectra of 15 proteins and one synthetic polypeptide. The digitized CD spectra of the reference proteins and polypeptide constitute the columns of the CD data matrix R and the fractions of... 

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