Cholesky decomposition

The Cholesky decomposition is an efficient and reliable way to check if a symmetric matrix is positive definite. If a symmetric matrix is not positive definite, the Cholesky decomposition will fail.

Theorem: If is symmetric positive definite (SPD), then has a unique Cholesky decomposition: where is upper triangular with positive diagonal entries.

Example

A = np.array([[7.3, 1, 0], [1, 20, 3.5], [0, 3.5, 2]])
print('----- Matrix A: -----\n' + str(A) + '\n')

# compute cholesky decomposition
R = cholesky_decomposition(A)

print('----- Matrix R: -----\n' + str(R) + '\n')
print('--> Verification: \n' + str(R.transpose() @ R))

Notes

1. G.H. Golub and C.F. Van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore, Maryland, 4th edition, 2013.

2. Å. Björck. Numerical Methods in Matrix Computations. Springer,2015.