Bidiagonal matrix

This matrix is an upper bidiagonal matrix with varying diagonal entries aj [y] and constant upper diagonal entries —V. 

In theory there is nothing wrong but for numerical computations, the above formulas are rather costly and inaccurate They involve inverting the upper triangular bidiagonal matrix Ax, whose inverse will be a dense upper triangular matrix. Moreover, formula (6.107) for Y involves the inverse of C which itself is given by a triple and complicated matrix product correction term subtracted from Ax in (6.105). 

Another solution of (10.5) called bidiagonalization is used in Partial Least Squares (PLS). Here the original input matrix X is linked to two orthogonal matrices O and W by a bidiagonal matrix L. 

Equation (11-5) is readily transformed into a bidiagonal matrix by use of the following procedure. Beginning with the bottom row, each row is added to the one above it and then the one above is replaced by the result so obtained. Originally,4 the component-material balances were formulated by enclosing the bottom of the column and stages s, s — 1, s — 2,. .., r + 3, r + 2, and r + 1. The equations so obtained are equivalent to those given by Eq. (11-14) see Prob. 11-8. 

One can show that the matrix P W is an upper bidiagonal matrix so that the PLS algorithm represents just a variation of diagonalizing a matrix before its inversion. 

Note that by commencing with the top row and adding each row to the one below it and then replacing the latter by the result so obtained, an upper triangular matrix which is bidiagonal is obtained (see Prob. 10-8). 

The matrix equation RV = —3F is readily solved by converting R to an upper triangular matrix which is bidiagonal by the same procedure described above for Af. 

The simplest method for PCA used in analytics is the iterative nonlinear iterative partial least squares (NIPALS) algorithm explained in Example 5.1. More powerful methods are based on matrix diagonalization, such as SVD, or bidiagonalization, such as the partial least squares (PLS) method. 

In Equation 6-41, the indexed quantities q, bj, q, Vj and Wj are called column vectors of dimension n sometimes, they are vectors denoted by the bold-faced symbols a, b, c, v and w. The square matrix at the left reveals its obvious tridiagonal structure it is a special case of a diagonal banded matrix. We will not deal with matrix inversion in this book. Suffice it to say that the last (or the first) equation, which involves two unknowns only, is usually used to reduce the number of unknowns along each row, right up (or down) the matrix, thus resulting in a two or bidiagonal system. Repeating the process in the opposite direction yields the solution vector v. 

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