對(duì)于正定矩陣 A，可對(duì)其進(jìn)行 Choleskey 分解，即：A=P'P，其中 P 為上三角矩陣，在

R 中可以用函數(shù) chol()進(jìn)行 Choleskey 分解，例如：

> A

[,1] [,2] [,3] [,4]

[1,] 2 1 1 1

[2,] 1 2 1 1

[3,] 1 1 2 1

[4,] 1 1 1 2

> chol(A)

[,1] [,2] [,3] [,4]

[1,] 1.414214 0.7071068 0.7071068 0.7071068

[2,] 0.000000 1.2247449 0.4082483 0.4082483

[3,] 0.000000 0.0000000 1.1547005 0.2886751

[4,] 0.000000 0.0000000 0.0000000 1.1180340

> t(chol(A))%*%chol(A)

[,1] [,2] [,3] [,4]

[1,] 2 1 1 1

[2,] 1 2 1 1

[3,] 1 1 2 1

[4,] 1 1 1 2

> crossprod(chol(A),chol(A))

[,1] [,2] [,3] [,4]

[1,] 2 1 1 1

[2,] 1 2 1 1

[3,] 1 1 2 1

[4,] 1 1 1 2

若矩陣為對(duì)稱正定矩陣，可以利用 Choleskey 分解求行列式的值，如：

> prod(diag(chol(A))^2)

[1] 5

> det(A)

[1] 5

若矩陣為對(duì)稱正定矩陣，可以利用Choleskey分解求矩陣的逆，這時(shí)用函數(shù)

chol2inv()，這種用法更有效。如：

> chol2inv(chol(A))

[,1] [,2] [,3] [,4]

[1,] 0.8 -0.2 -0.2 -0.2

[3,] -0.2 -0.2 0.8 -0.2

[4,] -0.2 -0.2 -0.2 0.8

> solve(A)

[,1] [,2] [,3] [,4]

[1,] 0.8 -0.2 -0.2 -0.2

[2,] -0.2 0.8 -0.2 -0.2

[3,] -0.2 -0.2 0.8 -0.2

[4,] -0.2 -0.2 -0.2 0.8

[2,] -0.2 0.8 -0.2 -0.2