What is the Cholesky factorisation of the matrix[4 2] [2 2]

The Cholesky factorization of a matrix is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It is useful for efficient numerical solutions, such as Monte Carlo simulations.

Given the matrix A:

A = [4 2]
    [2 2]

The Cholesky factorization of A is a lower triangular matrix L such that A = L * L^T.

Step 1: Compute the square root of the first element in the matrix, this will be the first element in L.

L[1,1] = sqrt(A[1,1]) = sqrt(4) = 2

Step 2: Divide the second element in the first row of A by the first element in L to get the second element in the first row of L.

L[1,2] = A[1,2] / L[1,1] = 2 / 2 = 1

Step 3: Subtract the square of the second element in the first row of L from the second element in the second row of A, then take the square root to get the second element in the second row of L.

L[2,2] = sqrt(A[2,2] - L[1,2]^2) = sqrt(2 - 1^2) = sqrt(1) = 1

Step 4: The second element in the first row of L is the same as the first element in the second row of L.

L[2,1] = L[1,2] = 1

So, the Cholesky factorization of A is:

L = [2 1]
    [1 1]

And indeed, L * L^T = A.