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Solving Linear Eqns 8/26/2003 page 1

Dennis L. BrickerDept of Mechanical & Industrial Engineering

University of Iowa

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Contents

Preliminarieso Elementary Matrix Operationso Elementary Matriceso Echelon Matriceso Rank of Matrices

Elimination Methodso Gauss Elimination

o Gauss-Jordan Elimination

Factorization Methodso Product Form of Inverse (PFI)o LU Factorizationo LDLTFactorizationo Cholesky LLTFactorization

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Solving Linear Eqns 8/26/2003 page 37

CHOLESKY FACTORIZATION

Suppose that A is a symmetric& positive definitematrix.

Then the Cholesky factorizationof A is

TA L L=

where L is a lower triangularmatrix.

Computation:Suppose that we have the factorization

TA L D L=

Then if 0ii

D , we can define a new diagonal matrix D where

i ii i

D D

Then ( ) ( ) T

T T T

A L D L L D D L LD LD L L= = = = where L LD=

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Example:

We wish to find the Cholesky factorization of the matrix

2 0 1

0 1 1

1 1 2

A

=

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Cholesky factorization

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The lower triangular matrix L is found by performing (on the

identity matrix) the inverse of the row operations used to reduce

the A matrix:

3 3 1

3 3 2

1 0 01

2 0 1 0

1 1 12

R R RL

R R R

+

= +

We now have the LU factorization of matrix A:

1 0 0 2 0 1

0 1 0 0 1 1

1 11 1 0 02 2

A LU

= =

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Define the diagonal matrix D:

1

1 0 02 0 0 2

0 1 0 0 1 0

1 0 0 20 02

D D

= =

Note that

1

1 0 0 2 0 12 0 1 0 0 1 1

0 0 2 10 02

11 02

0 1 10 0 1

U D U

= =

=

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And so,

11 01 0 0 2 0 0 2

0 1 0 0 1 0 0 1 1

1 1 0 0 11 1 0 02 2

TA LDL

= =

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Define the diagonal matrix D where i ii i

D D :

2 0 0

0 1 0

10 02

D

=

Then compute

1 0 0 2 0 0 2 0 0

0 1 0 0 1 0 0 1 0

1 1 1 11 1 0 0 12 2 2 2

L LD

= = =

So the Cholesky factorization is

12 02 0 0 2 0 1 0 0 1 1

1 1 11 0 02 2 2

TA LL

= =