Singular value decomposition theorem

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... 

An important theorem of matrix algebra, called singular value decomposition (SVD), states that any nxp table X can be written as the matrix product of three terms U, A and V ... 

In the previous section we have developed principal components analysis (PCA) from the fundamental theorem of singular value decomposition (SVD). In particular we have shown by means of eq. (31.1) how an nxp rectangular data matrix X can be decomposed into an nxr orthonormal matrix of row-latent vectors U, a pxr orthonormal matrix of column-latent vectors V and an rxr diagonal matrix of latent values A. Now we focus on the geometrical interpretation of this algebraic decomposition. 

To extract the linearly independent excitations, we shall have to use the so-called singular value decomposition of the valence density matrices generated by the creation/annihilation operators with valence-labels which axe present in the particular excitation operator in T. To illustrate this aspect, let us take an example. For any excitation operator containing the destruction of a pair of active orbitals from V o the overlap matrix of all such excited functions factorize, due to our new Wick s theorem, into antisymmetric products of one-body densities with non-valence labels and a two-particle density matrix ... 

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 ... 

In the derivation given by Golebiewski et al. (1979) the dioice of V was based on a maximum-overlap requirement. Here we simply note that any m X n matrix of rank n can be reduced to the above form by a so-called singular-value decomposition (see e.g. Amos and Hall, 1961 Stewart, 1973) the result is that is Hermitian while 2 is a zero matrix. The required theorem states that any m x n rectangular matrix M, of rank n, can be reduced as follows ... 

---