Cholesky Decomposition

A matrix in is:

• Lower triangular iff .
• Upper triangular iff .

1. Properties of P(S)DS Matrices

The following are some properties of Positive Defite Matrix :

• All its diagonal elements are positive: .
• . Hence, the largest coefficient of is on its diagonal.
• The matrices in the upper left corner of are also positive definite.

The following are some properties of Positive Semi-Definite Matrix :

• All its diagonal elements are non-negative: .
• . Hence, the largest coefficient of is on its diagonal.
• The matrices in the upper left corner of are also positive semi-definite.

2. Cholesky Decomposition

The cholesky decomposition of is a lower triangular matrix such that . It exists iff is positive semi-definite.

Additionally, if is positive definite, the diagonal elements of are positive.

It may be computed by brute force: setting each cell of as a variable, and solving the system of equations :

2.1 Using Cholesky

We can use cholesky decomposition to solve an equation , where :

1. Let . Solve by forward substitution.
2. Solve by back substitution to find .