Cholesky decomposition explanation simple terms

Could someone explain to me what Cholesky decomposition is in layman terms?

Like the simple intuition behind it. Thanks.

[Vidhi Agarwal]

Cholesky decomposition is a numerical method used to solve matrix systems of the form Ax=b, where A is a symmetric positive definite matrix, b is the column vector of known values, and x is the vector of unknown variables.

The Cholesky decomposition of a symmetric positive definite matrix A is given by the factorization:

A = LL^T

where L is a lower-triangular matrix and L^T is its transpose. This factorization method decomposes the original matrix into two triangular matrices, which are easier and faster to solve than the original matrix.

% Define the matrix A and the vector b for the system of linear equations Ax = b
A = [4 2 -1; 2 5 3; -1 3 13];
b = [5; -1; 24];

% Use the chol function to compute the Cholesky decomposition of A
L = chol(A,'lower');

% Use forward substitution to solve Ly = b
y = L \ b;

% Use backward substitution to solve L^Tx = y
x = L' \ y;

% Display the solution vector x
disp(x);

In this example, A is a 3x3 matrix representing the coefficients of the system of linear equations, and b is a 3x1 vector representing the constants. The chol function is used to compute the Cholesky decomposition of A, and the lower-triangular matrix L is stored in a variable.

The system of linear equations is then solved using two rounds of substitution: first, forward substitution is used to solve the equation Ly=b, and then backward substitution is used to solve the equation L^Tx=y. The solution vector x is then displayed using the disp() function.

Note that the chol function assumes that the matrix A is symmetric positive definite. If the matrix is not, the function will return an error. Also, Cholesky decomposition is useful only if the matrix A is symmetric positive definite because we can extract the lower triangular matrix using which we can then solve equations easily.