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Cholesky Decomposition Guide

Cholesky decomposition provides a method to factorize a symmetric positive definite matrix A into the product of a lower triangular matrix L and its transpose: A = LLT. This decomposition requires half the memory and about half as many operations as the general LU factorization. It is also more numerically stable since it does not require pivoting. Cholesky decomposition fails if the matrix is not positive definite. The decomposition can be used to efficiently solve systems of linear equations with a symmetric positive definite coefficient matrix.

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139 views9 pages

Cholesky Decomposition Guide

Cholesky decomposition provides a method to factorize a symmetric positive definite matrix A into the product of a lower triangular matrix L and its transpose: A = LLT. This decomposition requires half the memory and about half as many operations as the general LU factorization. It is also more numerically stable since it does not require pivoting. Cholesky decomposition fails if the matrix is not positive definite. The decomposition can be used to efficiently solve systems of linear equations with a symmetric positive definite coefficient matrix.

Uploaded by

sirj0_hn
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We take content rights seriously. If you suspect this is your content, claim it here.
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Cholesky factorization

Positive de nite matrices

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Cholesky decomposition

Let a real matrix A is


symmetric: AT = A
positive de nite: xT Ax > 0 ∀x ∈ Rm

Then

A = LLT
where L is a lower triangular matrix.

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Cholesky decomposition

A = LLT
  
l11 0 0 ··· 0 l11 l21 l31 ··· lm1
 l21 l22 0 ··· 0   0 l22 l32 ··· lm2 
  
 ···  0 ··· lm3 
=  l31 l32 l33 0  0 l33 
 .. .. .. ..   .. .. .. .. 
 . . . .  . . . . 
lm1 lm2 lm3 · · · lmm 0 0 0 ··· lmm

1st row of A
2
a11 = l11
a12 = l11 l21 , ··· , a1k = l11 lk1 , k = 2, . . . , m

NB: The square root is OK because A is positive de nite.

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Cholesky decomposition

A = LLT
  
l11 0 0 ··· 0 l11 l21 l31 ··· lm1
 l21 l22 0 ··· 0   0 l22 l32 ··· lm2 
  
 ···  0 ··· lm3 
=  l31 l32 l33 0  0 l33 
 .. .. .. ..   .. .. .. .. 
 . . . .  . . . . 
lm1 lm2 lm3 · · · lmm 0 0 0 ··· lmm

2nd row of A
a21 = l21 l11
2
a22 = l21 2
+ l22 , ··· , a2k = l21 lk1 + l22 lk2 , k = 2, . . . , m

NB: The square root is OK because A is positive de nite.

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Cholesky decomposition

Compared to a general LU factorization, Cholesky decomposition:


▶ requires 1/2 memory
▶ requires ∼ 1/2 less operations
▶ has better stability, and does not require pivoting
▶ fails if A is not PD.

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Systems of linear equations

If A is symmetric and positive de nite, then

Ax = b

is solved via A = LLT , and

Ly = b (forward substitution)
LT x = y (back substitution)

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Applications of Cholesky decomposition

Quantum mechanics ( )∗
Observables are represented by Hermitian operators. ( AT = A)

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Applications of Cholesky decomposition

Numerical optimization
The Hessian matrix of a multivariate function F (x)

∂2F
Hjk =
∂xj ∂xk

is symmetric and is in some cases positive (semi-)de nite.

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Applications of Cholesky decomposition

Monte Carlo simulations


Generation of correlated Gaussian random variables: decompose
the correlation matrix C = LLT , generate a vector of uncorrelated
values x, then
z = Lx
has the correlation matrix C.

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