Moore-Penrose-matrix

Computation of the regression coefficients b vectorwise is carried out by formation of the pseudo-inverse matrix X" " (Moore-Penrose matrix) according to... 

In the case of full rank, all singular values will be obviously different from zero and the SVD solution equals that of OLS. However, one often comes up with several small singular values because of ill-conditioned systems. Therefore, the main goal of PCR is not to keep all singular values for an exact representation of the Moore-Penrose matrix, but to select a subset of singular values that best guarantee predictions of unknown cases. 

It should be noted that in the case of a singular matrix A, the dimensions of V and A are pxr and rxr, respectively, where r is smaller than p. The expression in eq. (29.53) allows us to compute the generalized inverse, specifically the Moore-Penrose inverse, of a symmetric matrix A from the expression ... 

In algebra, a number multiplied by its inverse results in a value of 1. In matrix algebra, the inverse of a square matrix (denoted by a superscript T) multiplied by itself results in the identity matrix. In other words, the inverse of X is the matrix X-1 such that XX-1 = X-1X = I. Two matrices are said to be orthogonal or independent if XYT = I. The inverse of an orthogonal matrix is its transpose. Not all matrices can be inverted. However, one condition for inversion is that the matrix must be square. Sometimes an inverse to a matrix cannot be found, particularly if the matrix has a number of linearly dependent column. In such a case, a generalized estimate of inverted matrix can be estimated using a Moore Penrose inverse (denoted as superscript e.g., X-). 

In equations 7.13,7.25, and 7.27 we have denoted L+ = (L Ly L1, and similarly for Mi+ in equation 7.27, and D+ in equation 7.28, and N4- in equation 7.29. This notation is used because these are all special cases of the Moor e-Penrose Pseudoinverse M which can be defined for an arbitrary matrix M and which gives the minimum least-squares approximation even in cases where the columns of M may not be linearly independent (see Lawson and Hanson, 1974). Similarly, in equations 7.25, 7.27, and 7.29 we have denoted Ai+ = A/ (AiA/) 1 since this is another special case of the Moore-Penrose Pseudoinverse, for the case where the matrix in question, Ai, has linearly independent rows. 

The determination of output weights between hidden and output layers is to find the least-square solution to the given linear system. The minimum norm least-square solution to hnear system (1) is M Y, where M is the Moore-Penrose generalized inverse of matrix M. The minimum norm least-square solution is unique and has the smallest norm among the least-square solutions. 

Since C is not a square matrix (it is 4 X 3), the unknown X, Y, Z coordinates can be solved by using the Moore-Penrose generalized inverse, as follows ... 

To calculate the correction vector, Eq. 6.19 is solved via matrix inversion, e.g. by calculating the Moore-Penrose generalized matrix inverse ... 

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