91Ó°ÊÓ

Theorem 12 in section 6.4states that when \(A\) is a \(m \times n\) matrix that islinearly independent columns, then \(A\) may be factored as \(A = QR\), with \(Q\) is a \(m \times n\) matrix wherein columns provide an orthonormal basisfor \({\mathop{\rm Col}\nolimits} A\), and \(R\) is an \(n \times n\) upper triangular invertible matrix which has positive entries on its diagonal.

Assume that \(A\) is positive definite, and suppose that Cholesky factorization of \(A = {R^T}R\) where \(R\) is an upper triangular and having positive diagonal entries. Consider that \(D\) as the diagonal matrix with diagonal entries equal to diagonal entries of \(R\). The matrix \(L = {R^T}{D^{ - 1}}\) is lower triangular wherein 1’s on its diagonal because right-multiply by a diagonal matrix scales the columns of the matrix on its left.

When \(U = DR\) then \(A = {R^T}{D^{ - 1}}DR = LU\).

Conversely, consider that A contains a LU factorization, that is \(A = LU\), where \(U\) contains positive pivots on its diagonal. Consider \(D\) as the diagonal matrix and \(\sqrt {{u_{11}}} , \ldots ,\sqrt {{u_{nn}}} \) on its diagonal. The matrix \(V = {\left( {{D^2}} \right)^{ - 1}}U\) is upper triangular wherein 1’s on its diagonal because multiply by a diagonal matrix on its left scales the rows of the matrix on its right and we obtain \(A = L{D^2}V\). As \(A\) is symmetric, then \({A^T} = {\left( {L{D^2}V} \right)^T} = {V^T}{D^2}{L^T} = A\), and \(L = {V^T}\). When \(R = DV = {D^{ - 1}}U\) then \(A = {V^T}DDV = {\left( {DV} \right)^T}\left( {DV} \right) = {R^T}R\).

Hence, it is proved that A has a LU factorization \(A = LU\).