Solve \(Ax=b\) with \(A=\begin{pmatrix}4&2\\2&5\end{pmatrix}\) and \(b=\begin{pmatrix}2\\1\end{pmatrix}\) using the Cholesky factor \(L\). What is \(x\)?
Solving \(Ly=b\) gives \(y=(1,0)^{T}\), and then \(L^{T}x=y\) gives \(x=(1/2,0)^{T}\). Checking \(Ax=b\) confirms the result; the option equal to \(b\) skips the solving step.
The Cholesky factorization decomposes a symmetric positive definite matrix \(A\) as \(A = LL^{T}\), where \(L\) is lower triangular with positive diagonal.