Steps for Cholesky Decomposition

To decompose or factorize any Hermitian symmetric matrix, we can use the following steps:

Step 1: First write the given matrix in the decomposed form.

Let A be the positive definite symmetric matrix which can be decomposed as A = LL^*.

Step 2: Now, we have to evaluate matrix L where L is defined as:

Step 3: To evaluate the diagonal element, use the formula:

L_vv = √(A_vv – ∑_u<v L_vu (L_vu)*)

Where L* is the conjugate of L.

Step 4: To evaluate the non-diagonal element, use the formula:

L_tv = (1 / L_vv)(A_tv – ∑_u<v L_tu (L_vu)*)

Where L* is the conjugate of L.

Step 5: Put all the elements in L and evaluate A = LL^*.

Example: Find the Cholesky decomposition for a matrix A =

Solution:

A =

According to Cholesky decomposition the matrix can be written as:

A = LL*

where L is the lower triangular matrix of A and L* is the conjugate transpose matrix of L.

First, we have to find the lower triangular matrix of A.

Let the lower triangular matrix for P be L =

We calculate the elements of lower triangular matrix using the following formulas:

The diagonal element of L is calculated using:

L_vv = √(A_vv – ∑_u<v L_vu (L_vu)*)

The non-diagonal element is calculated using formula:

L_tv = (1 / L_vv)(A_tv – ∑_u<v L_tu (L_vu)*)

L₁₁ = √(A₁₁ – ∑_u<1 L_1u (L_1u)*) [Since, there is no value u which is lesser than 1(here, v=1) so, ∑_u<v L_vu (L_vu)*= 0 ]

⇒ L11 = √(81 – 0) = √81 = 9

L₂₁ = (1 / L₁₁)(A₂₁ )

⇒ L₂₁ = (1 / 9)(9i) = i

(L₂₁)* = -i (conjugate of i)

L22 = Ltv = (1 / Lvv)(Atv – ∑u<v Ltu (Lvu)*)

(∑_u<v L_tu (L_vu)* is the summation of the values of L where u<v)

L₂₂ = √(A₂₂ – ∑_u<2 L₂₁ (L₂₁)*) Here, v = 2 so u should be the values less than 2. By putting values in formula)

⇒ L₂₂ = √(A₂₂ – L₂₁ (L₂₁)*)

⇒ L₂₂ = √(45 – (-i) (i))

⇒ L₂₂ = √(45 + i²)

⇒ L₂₂ = √(45 – 1) [i² = -1]

⇒ L₂₂ = √44

The lower triangular matrix of P according to Cholesky decomposition is:

L =

Now, we will find L* i.e., conjugate transpose of L

L* =

Finally, we apply

A = LL*

A =

The final matrix A after applying Cholesky decomposition is:

A =

Cholesky Decomposition is one of the types of many decompositions in linear algebra which is a branch of mathematics that deals with linear equations and vectors. Decomposition is the term related to the factorization of matrices in linear algebra, and Cholesky is one of the ways to factorize or decompose the matrix into two matrices. This article explores the Cholesky Decomposition in detail including its definition, steps to factorize matrices using Cholesky Decomposition, and some of the solved examples. So, let’s start learning about this exciting topic of Cholesky Decomposition.