Definition:Cholesky Factorization

Definition

Let $\mathbf A$ be a positive definite matrix.

A Cholesky factorization of $\mathbf A$ is an expression of the form:

$\mathbf A = \mathbf R^\intercal \mathbf R$

where:

$\mathbf R$ is an upper triangular matrix with diagonal entries which are (strictly) positive

$\mathbf R^\intercal$ denotes the transpose of $\mathbf R$.

Also defined as

Some sources in their definition of a Cholesky factorization of a positive definite matrix $\mathbf A$ further specify:

$(1): \quad$ that $\mathbf A$ is square

$(2): \quad$ that $\mathbf A$ is symmetric

but both these properties follow a fortiori from the definition of a positive definite matrix.

Also known as

A Cholesky factorization is also known as a Cholesky decomposition.

Examples

Arbitrary Example 1

Arbitrary simple example of a positive definite matrix with its Cholesky factorization:

$\begin {pmatrix} 1 & -1 \\ -1 & 5 \end {pmatrix} = \begin {pmatrix} 1 & 0 \\ -1 & 2 \end {pmatrix} \begin {pmatrix} 1 & -1 \\ 0 & 2 \end {pmatrix}$

Also see

• Positive Definite Matrix has Cholesky Factorization

• Definition:Cholesky Factor

• Results about Cholesky factorizations can be found here.

Source of Name

This entry was named for André-Louis Cholesky.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cholesky factorization
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): factorization: 2. (of a matrix)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cholesky factorization
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): factorization: 2. (of a matrix)