Cholesky Decomposition

Definition

$A=L{L}^{†}$

Decomposition of a Positive-Definite Matrix into the product of lower triangular matrix and its Conjugate Transpose.

Computation

Let $\mathbf{A}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{22}& a_{33}\end{array}\right]$ be Positive-Definite Matrix. Then, $\mathbf{A}=\mathbf{L}{\mathbf{L}}^{⊺}=\left[\begin{array}{ccc}{L}_{11}& 0& 0\\ L_{21}& L_{22}& 0\\ L_{31}& L_{32}& L_{33}\end{array}\right]\left[\begin{array}{ccc}{L}_{11}& L_{21}& L_{31}\\ 0& L_{22}& L_{32}\\ 0& 0& L_{33}\end{array}\right]=\left[\begin{array}{ccc}{L}_{11}^2& & \text{(symmetric)}\\ & & \\ L_{21}{L}_{11}& L_{21}^2+L_{22}^2& \\ & & \\ L_{31}{L}_{11}& L_{31}{L}_{21}+L_{32}{L}_{22}& L_{31}^2+L_{32}^2+L_{33}^2\end{array}\right]$ By setting the first-row first-column entry $L_{11}=\sqrt{a_{11}}$, can find other entries using substitution. $L_{21}=a_{21}/L_{11},L_{31}=a_{31}/L_{11},L_{22}=\sqrt{a_{22}-L_{21}^2},L_{32}=(a_{32}=L_{31}{L}_{21})/L_{22},L_{33}=\sqrt{a_{33}-L_{31}^2-L_{32}^2}$ By summarizing, $L_{ii}=\sqrt{a_{ii}-{\sum }_{k=1}^{i-1}{L}_{ik}^2}$, $L_{ij}=\frac{1}{L_{jj}}(a_{ij}-{\sum }_{k=1}^{j-1}{L}_{ik}{L}_{jk})$