7.6. Cholesky Decomposition

Let’s choose a symmetric positive definite matrix:

> A <- matrix(c(4, 12, -16, 12, 37, -43, -16, -43, 98), nrow=3)
> A

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

[1,] 4 12 -16 [2,] 12 37 -43 [3,] -16 -43 98

Let’s perform its Cholesky decomposition:

> U <- chol(A)
> U
     [,1] [,2] [,3]
[1,]    2    6   -8
[2,]    0    1    5
[3,]    0    0    3

Let’s verify the correctness of the decomposition:

> t(U) %*% U
     [,1] [,2] [,3]
[1,]    4   12  -16
[2,]   12   37  -43
[3,]  -16  -43   98

Alternative way:

> crossprod(U)
     [,1] [,2] [,3]
[1,]    4   12  -16
[2,]   12   37  -43
[3,]  -16  -43   98