Cholesky Decomposition Examples
Example 1: Find the Cholesky decomposition for a matrix X whose lower triangular matrix is given by: L =
Solution:
The lower triangular matrix is given:
L =
The conjugate transpose of the above lower triangular matrix is:
L* =
From Cholesky decomposition X can be written as:
X = L L*
X =
X =
Example 2: Find the lower triangular matrix using Cholesky decomposition for a matrix A =
Solution:
Let the lower triangular matrix for A be L =
The diagonal element of L is calculated using:
Lvv = β(Avv β βu<v Lvu (Lvu)*)
The non-diagonal element is calculated using formula:
Ltv = (1 / Lvv)(Atv β βu<v Ltu (Lvu)*)
L11 = β(A11 β βu<1 L1u (L1u)*) [βu<1 L1u (L1u)* = 0 as v = 1 is the least value and there is no u lesser than v]
β L11 = β(36 β 0)
β L11 = 6
L21 = (1 / L11)(A21 β βu<1 L2u (L1u)*)
[Here, there is no u which is lesser than 1 so, βu<1 L2u (L1u)* = 0]
β L21 = (1 / L11)(A21 β 0)
β L21 = (1 / 6)(-12i) = -2i
Thus, (L21)* = 2i
L22 = β(A22 β βu<2 L21 (L21)*) [Here, only u = 1 is the value satisfying the condition u < 2. Putting the values in the formula]
β L22 = β(A22 β L21 (L21)*)
β L22 = β(42 β (-2i) (2i))
β L22 = β(42 + 4i2)
β L22 = β(42 β 4) [i2 = -1]
β L22 = β38
The lower triangular matrix of A according to Cholesky decomposition is:
L =
Example 3: Find the Cholesky decomposition for a matrix X whose lower triangular matrix is given by: L =
Solution:
The lower triangular matrix is given:
L =
The conjugate transpose of the above lower triangular matrix is:
L* =
From Cholesky decomposition X can be written as:
X = L L*
X =[Tex] \begin {bmatrix} 1&2&5\\ 0& -3i&-i\\ 0& 0 &-2i \end{bmatrix} [/Tex]
X =
Cholesky Decomposition
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, letβs start learning about this exciting topic of Cholesky Decomposition.