使用chol()分解正定矩阵(positive definite matrix), Choleskey

一个n×n的实对称矩阵是正定的，当且仅当对于所有的非零实系数向量z，都有z^Tz > 0。其中z^T表示z的转置。

对于复数的情况，定义则为：一个n×n的埃尔米特矩阵(或厄米矩阵)是正定的当且仅当对于每个非零的複向量z，都有z^*z > 0。其中z^*表示z的共轭转置。由于是埃尔米特矩阵，经计算可知，对于任意的複向量z，z^*z必然是实数，从而可以与0比较大小。因此这个定义是自洽的。

> a <- diag(3)+1
> a
     [,1] [,2] [,3]
[1,]    2    1    1
[2,]    1    2    1
[3,]    1    1    2
> chol(a)
         [,1]      [,2]      [,3]
[1,] 1.414214 0.7071068 0.7071068
[2,] 0.000000 1.2247449 0.4082483
[3,] 0.000000 0.0000000 1.1547005
> t(chol(a))
          [,1]      [,2]     [,3]
[1,] 1.4142136 0.0000000 0.000000
[2,] 0.7071068 1.2247449 0.000000
[3,] 0.7071068 0.4082483 1.154701

> crossprod(chol(a))
     [,1] [,2] [,3]
[1,]    2    1    1
[2,]    1    2    1
[3,]    1    1    2
> t(chol(a)) %*% chol(a)
     [,1] [,2] [,3]
[1,]    2    1    1
[2,]    1    2    1
[3,]    1    1    2

> prod(diag(chol(A))^2)
[1] 5
> det(A)
[1] 5

> chol2inv(chol(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
> 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