Cholesky factorization for $2 \times 2$ matrices. Show that any positive definite $2 \times 2$ matrix A can be written uniquely $A = L L^{T}$ as where L is a lower triangular $2 \times 2$ matrix with positive entries on the diagonal. Hint: Solve the equation
$[\begin{matrix} a & b \\ b & c \end{matrix}] = [\begin{matrix} x & 0 \\ y & z \end{matrix}] [\begin{matrix} \begin{matrix} x & y \\ 0 & z \end{matrix} \end{matrix}]$

Question

Accepted Answer

Every positive definite 2 x 2 matrix

$A = [\begin{matrix} a & b \\ b & c \end{matrix}]$

has the form $A = L L^{T}$ in Cholesky factorization, where,

$L = [\begin{matrix} \sqrt{a} & 0 \\ b / \sqrt{a} & \sqrt{a c - b^{2}} & / \sqrt{a} \end{matrix}] \begin{matrix} \end{matrix}$

is a positive diagonal matrix with lower triangular entries

Short Answer