(Cholesky Decomposition) This problem outlines the basic ideas of a popular and effectiv...

(Cholesky Decomposition) This problem outlines the basic ideas of a popular and effective method of computing least squares estimates. Assuming that its inverse exists, X^TX is a positive, definite matrix and may be factored as X^TX = R^T 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

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.