(Cholesky Decomposition) This problem outlines the basic

Chapter , Problem 9

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(Cholesky Decomposition) This problem outlines the basic ideas of a popular and effective method of computing least squares estimates. Assuming that its inverse exists, \(\mathbf{X}^T \mathbf{X}\) is a positive, definite matrix and may be factored as \(\mathbf{X}^T \mathbf{X}=\mathbf{R}^T \mathbf{R}\), where R is an upper-triangular matrix. This factorization is called the Cholesky decomposition. Show that the least squares estimates can be found by solving the equations

                                          \(\begin{aligned}\mathbf{R}^T \mathbf{v} & =\mathbf{X}^T \mathbf{Y} \\\mathbf{R} \hat{\boldsymbol{\beta}} & =\mathbf{v}\end{aligned}\)

where v is appropriately defined. Show that these equations can be solved by back-substitution because R is upper-triangular, and that therefore it is not necessary to carry out any matrix inversions explicitly to find the least squares estimates.