LU Factorization

333231

2221

11

0

00

LLL

LL

L

L

LUA

33

2322

131211

00

0

U

UU

UUU

U

333231

232221

131211

AAA

AAA

AAA

A

333231

2221

11

0

00

LLL

LL

L

33

2322

131211

00

0

U

UU

UUU

333231

232221

131211

AAA

AAA

AAA

333323321331223212311131

23221321222212211121

131112111111

ULULULULULUL

ULULULULUL

ULULUL

333231

232221

131211

AAA

AAA

AAA

311131 AUL 3222321231 AULUL

33333323321331 AULULUL

111111 AUL 121211 AUL 131311 AUL

2323221321 AULUL 211121 AUL 2222221221 AULUL

Equating the elements of the First Row :-

Equating the elements of the 2nd Row :-

Equating the elements of the 3rd Row :-

We have 12 unknowns but only 9 equations. We need some sort of compromise.

Crout’s Method

Set 1332211 UUU

Dolittle’s Method

Set 1332211 LLL

Crout's Method

U1 1 1 U

2 2 1 U3 3 1

L1 1

A1 1

U1 1

U1 2

A1 2

L1 1

U1 3

A1 3

L1 1

L2 1

A2 1

U1 1

L2 2

A2 2 L

2 1 U1 2

U2 2

U2 3

A2 3 L

2 1 U1 3

L2 2

L3 1

A3 1

U1 1

L3 2

A3 2 L

3 1 U1 2

U2 2

L3 3

A3 3 L

3 1 U1 3 L

3 2 U2 3

U3 3

bLy

bLUx

bAx

Solve for y, and then solve for x.

yUx

Use of LU factors in solving systems of linear equations

A

2

1

3

1

1

2

3

2

4

B

13

7

5

L

2

1

3

0

0.5

3.5

0

0

25

U

1

0

0

0.5

1

0

1.5

7

1

Y1

B1

L1 1

Y2

B2

L2 1 Y

1

L2 2

5

7

13

25533

0501

002

3

2

1

Y

Y

Y

.

.

Y3

B3

L3 1 Y

1 L

3 2 Y2

L3 3

Y

6.5

1

0.84

LUX = B LY = B

840

1

56

100

710

51501

3

2

1

.

...

X

X

X

UX = Y

X3

Y3

U3 3

X2

Y2

U2 3 X

3

U2 2

X1

Y1

U1 2 X

2 U

1 3 X3

U1 1

X

1.8

6.88

0.84

Elementary Matrices and The LU Factorization

Definition: Any matrix obtained by performing a single elementary row operation (ERO) on the identity (unit) matrix is called an elementary matrix.

There are three elementary operations:Permute rows i and j Multiply row i by a non-zero scalar k Add k times row i to row j

Corresponding to the three ERO, we have then three elementary matrices:

Type 1: - permute rows i and j in In. Type 2: - multiply row i of In by a non-zero scalar k Type 3: - Add k times row i of In to row j

100

001

010

12P

100

00

001

2 kkM

100

01

001

12 kkA

Permutation matrix:

Scaling matrix:

Row combination:

Pre-multiplying a matrix A by an elementary matrix E has the effect of performing the corresponding ERO on A.

Example: We can multiply the First row of the matrix A by 3 (an elementary row operation). The resulting matrix will become

574

132A

574

396

We can achieve the same result by pre-multiplying A by the corresponding elementary matrix.

574

396

574

132

10

0331 AM

An ERO can be performed on a matrix by pre-multiplying the matrix by a corresponding elementary matrix. Therefore, we can show that any matrix A can be reduced to a row echelon form (REF) by multiplication by a sequence of elementary matrices.

RAEEE k 21

where R denotes an REF of A.

nk IAEEE 21

Since the unique reduced row echelon form (RREF) of a matrix is the unit matrix

nk IAEEE 21

nIAA 1

kEEEA 211

nk IEEEA 211

A nonsingular matrix can be reduced to an upper triangular matrix using elementary row operations of Type 3 only. The elementary matrices corresponding to Type 3 EROs are unit lower triangular matrices. We can write

UAEEE k 21

Since each elementary matrix is nonsingular (meaning their inverse exist) we can write

UEEEEA kk1

11

21

11

UAEEE k 21

We know that the product of two lower triangular matrices is also a lower triangular matrix. Therefore

LUA 1

11

21

11

EEEEL kk

Inverses of the three elementary matrices are:

kMkM ii 11

ijij PP 1

)( kAkA ijij 1

Determine the LU factorization of the matrix

121

213

352

A

First, let us do the EROs to reduce A into an upper triangular matrix.

25290

2132130

352

2123

121

213

352

1312 A,A

13923A

200

2132130

352

These EROs can be written in terms of their equivalent elementary matrices as

200

2132130

352

321 AEEE

2321139 123132231 AE,AE,AE

200

2132130

352

U

11

12

13

EEEL

2321139 121

3131

2231

1 AE,AE,AE

11390

010

001

1021

010

001

100

0123

001

L

113921

0123

001

L

200

2132130

352

113921

0123

001

A

We can construct the lower triangular matrix L without multiplying the elementary matrices if we utilize the multipliers obtained while we converted matrix A into an upper triangular matrix.

2321139 123132231 AE,AE,AE

Definition: When using ERO of Type 3, the multiple of a specific row i that is subtracted from row j to put a zero in the ji position is called a multiplier, and is denoted as jim

1392123 323121 m,m,m

113921

0123

001

L

1392123 323121 m,m,m

If we notice the unit lower triangular matrix L carefully, we see that the elements beneath the leading diagonal are just the corresponding multipliers. This relationship holds in general. Therefore, we can do elementary row operations of Type 3 to reduce A to upper triangular form and then utilize the corresponding multipliers to write L directly.