LU decomposition

LU decomposition

Definition 1

Let

A

be an

n

×

m

matrix. An

LU decomposition

(sometimes also called an LU factorization) of

A

, if it
exists, is an

n

×

n

unit lower triangular matrix

L

and an

n

×

m

matrix U, in (upper)

echelon form

, such that

A

=

L

⁢

U

The LU factorization is closely related to the

row reduction

algorithm

. In a very real sense, the factorization is a record of the
steps taken in row reducing a matrix to echelon form. The matrix

L

“encodes” the

sequence

of row replacement

operations

that row
reduce the given matrix

A

to echelon form

U

.

Proposition 2

Suppose that

A

=

L

⁢

U

is an LU factorization, and let

L

i

⁢

j

denote
the entries of

L

. Then, the row reduction

A

⟹

ρ

U

is accomplished by the
following sequence

ρ

of

1

2

⁢

n

⁢

(

n

-

1

)

row replacement
operations:

row

j

↦

row

j

-

L

j

⁢

1

⁢

row

1

,

j

=

2

,

…

,

n

;

row

j

↦

row

j

-

L

j

⁢

2

⁢

row

2

,

j

=

3

,

…

,

n

;

⁢

⋮

row

j

↦

row

j

-

L

j

⁢

k

⁢

row

k

,

j

=

k

+

1

,

…

,

n

,

k

=

1

,

…

,

n

-

1

⁢

⋮

Note: the first

n

-

1

steps clear out column

1

, the next

n

-

2

steps
clear out column

2

, etc.

Proposition 3

Not every matrix admits an LU factorization. Indeed, an LU
factorization exists if and only if

A

can be

reduced

to echelon
without using row exchange operations. However, if an LU
factorization exists, then it is unique.

In the most general case, one has to employ row exchange operations.

Proposition 4

Let

A

be an

n

×

m

matrix. Then, there exists an

n

×

n

permutation matrix

P

(indeed, many such) such that the matrix

P

⁢

A

admits an LU factorization, i.e., there exist matrices

P

,

L

,

U

such that

P

⁢

A

=

L

⁢

U

.

The key idea behind LU factorization is that one does not need to
employ row scalings to do row reduction until the second half (the
back-substitution phase) of the algorithm. This has significant

implication

for

numerical stability

of the algorithm.

The LU decomposition of a given matrix

A

is useful for the solution
of the

systems of linear equations

of the form

A

⁢

x

=

b

. Indeed, it
suffices to first solve the

linear system

L

⁢

y

=

b

, and second, to solve
the system

U

⁢

x

=

y

. This two-step procedure is easy to implement,
because owing to the lower and upper-triangular nature of the
matrices

L

and

U

, the required row operations are determined, more
or less, directly from the entries of

L

and

U

. Indeed, the first
step,

[

L

⁢

b

]

⟹

ρ

1

[

I

⁢

y

]

a sequence of row operations

ρ

1

, and the second step

[

U

⁢

y

]

⟹

ρ

2

[

E

⁢

x

0

]

a sequence of row operations

ρ

2

, are exactly the same row operations
one has to perform to row reduce

A

directly to

reduced echelon form

E

:

[

A

⁢

b

]

⟹

ρ

1

[

U

⁢

y

]

⟹

ρ

2

[

E

⁢

x

0

]

.

Note:

x

0

is the particular solution of

A

⁢

x

=

b

that sits in the
rightmost colulmn of the augmented matrix at the termination of the
row-reduction algorithm.

Title

LU decomposition

Canonical

name

LUDecomposition

Date of creation

2013-03-22 12:06:37

Last modified on

2013-03-22 12:06:37

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

10

Author

rmilson (146)

Entry type

Definition

Classification

msc 65-00

Classification

msc 15-00

Synonym

LU factorization

Synonym

LU-factorization

Synonym

LU-decomposition

Related topic

QRDecomposition

Generated on Sat Feb 10 12:34:42 2018 by

LaTeXML