Linear algebra Singular Value Decomposition

In Chapters 21-23 and in this chapter, we have described the most basic calculations for MLR, PCR, and PLS. To reiterate, our intention is to demonstrate these basic computations for each mathematical method presently, and then to delve into greater detail as the chapters progress consider these articles linear algebra bytes. For this chapter we will illustrate the basic calculation and mathematical relationships of different matrices for the calculations of Singular Value Decomposition or SVD. 

We will describe the PCA method following the treatment of Ressler et al. (2000) but with the above notation. PCA can be derived from the singular-value decomposition theorem from linear algebra, which says that any rectangular matrix can be decomposed as follows... 

The singular value decomposition (SVD) method, and the similar principal component analysis method, are powerful computational tools for parametric sensitivity analysis of the collective effects of a group of model parameters on a group of simulated properties. The SVD method is based on an elegant theorem of linear algebra. The theorem states that one can represent an w X n matrix M by a product of three matrices ... 

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