第2197题：Cholesky factorization

Andre-Louis Cholesky (1875–1918)

The following conditions are equivalent for a symmetric $n \times n$ matrix $A$ ：

1. $A$ is positive definite.

2. det ( ${(r)} \atop A$ ) $>0$ for each $r=1,2,\cdots,n$ .

3. $A=U^T U$ where $U$ is an upper triangular matrix with positive entries on the main diagonal.

Furthermore, the factorization in $A=U^T U$ is unique, called the Cholesky factorization of $A$ .

If $A$ is a positive definite matrix, the Cholesky factorization $A=U^T U$ can be obtained as follows:

Step 1. Carry $A$ to an upper triangular matrix $U_1$ with positive diagonal entries using row operations each of which adds a multiple of a row to lower row.

Step 2. Obtain $U$ from $U_1$ by dividing each row of $U_1$ by the square root of the diagonal entry in that row.

OK, let's find the Cholesky factorization $U$ of $A= \begin{bmatrix} 3 & 0 & 1 \\ 0 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}$ .

A. $U= \begin{bmatrix} 3 & 0 & 1 \\ 0 & 2 & 1 \\ 0 & 0 & \dfrac{7}{6} \end{bmatrix}$

B. $U= \begin{bmatrix} 3 & 0 & \dfrac{1}{\sqrt{3}} \\ 0 & 2 & \dfrac{1}{\sqrt{2}} \\ 0 & 0 & \dfrac{\sqrt{7}}{\sqrt{6}} \end{bmatrix}$

C. $U= \begin{bmatrix} \dfrac{1}{\sqrt{3}} & 0 & \sqrt{3} \\ 0 & \dfrac{1}{\sqrt{2}}& \sqrt{2} \\ 0 & 0 & \dfrac{\sqrt{7}}{\sqrt{6}} \end{bmatrix}$

D. $U= \begin{bmatrix} \sqrt{3} & 0 & \dfrac{1}{\sqrt{3}} \\ 0 & \sqrt{2} & \dfrac{1}{\sqrt{2}} \\ 0 & 0 & \dfrac{\sqrt{7}}{\sqrt{6}} \end{bmatrix}$

Andre-Louis Cholesky (1875–1918), was a French mathematician who died in World War I. His factorization was published in 1924 by a fellow officer.