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Solving Linear Eqns 8/26/2003 page 1
Dennis L. BrickerDept of Mechanical & Industrial Engineering
University of Iowa
Solving Linear Eqns 8/26/2003 page 2
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=
Solving Linear Eqns 8/26/2003 page 38
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
Solving Linear Eqns 8/26/2003 page 40
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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Solving Linear Eqns 8/26/2003 page 41
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
= =
=
Solving Linear Eqns 8/26/2003 page 42
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
= =