System with positive definite matrix
Task number: 2728
Use Cholesky factorization of \(\mathbf A\) to solve system \(\mathbf A\mathbf x = (10, 21, -32, 26, 23)^T\), where
\( \mathbf A=\begin{pmatrix} 1& 2& -3& 2& 1\\ 2& 5& -6& 3& 2\\ -3& -6& 10& -5& -3\\ 2& 3& -5& 15& 11\\ 1& 2& -3& 11& 14 \end{pmatrix} \)
Hint
For \(\mathbf A=\mathbf U^H\mathbf U\mathbf x\) use the substitution \(\mathbf U\mathbf x=\mathbf y\).
Resolution
The decomposition is \( \mathbf U= \begin{pmatrix} 1& 2&-3& 2& 1\\ 0& 1& 0&-1& 0\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 3& 3\\ 0& 0& 0& 0& 2 \end{pmatrix} \).
The system \(\mathbf U^H\mathbf y=\mathbf b\) yields \(\mathbf y=(10, 1,-2, 3, 2)^T\) and \(\mathbf U\mathbf x=\mathbf y\) yields \(\mathbf x=(1, 1, -2, 0, 1)^T\).
Since both matrices are in the echelon form, it suffices to use twice the backward subtitution.
Result
The system has a solution \( \mathbf x=(1, 1, -2, 0, 1)^T \).
