Lecture 15: LU Factorization
Daniel Frances c 2012
Contents
1
LU Factorization
1
1.1
Permutations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Can any nonsingular matrix be LU factorized?
. . . . . . . . . . . . . . . .
3
1.3
Recursive LU Factorization Process
. . . . . . . . . . . . . . . . . . . . . . .
4
2
An example
4
2.1
first 3x3 step in the recursion
. . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
next 2x2 step in the recursion
. . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Putting it altogether
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.4
Checking it out
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3
O
n
Analysis
5
4
Comparing the various factorizations
6
5
Supplementary Section: The divide by zero problem
7
1
LU Factorization
This is the last method we are covering for matrix inversion. It works for any nonsingular
matrix, and competes directly with Gaussian elimination. It turns out it is not as efficient
as Gaussian elimination, but it has a feature which makes LU preferable, not for matrix
inversion ‘per se’, but for solving linear equations  a close cousin to matrix inversion  which
we will deal with briefly a bit later.
In the LU factorization approach, as with Cholesky, we first factorize a matrix as a product
of a lower and upper triangular matrices and then use the result to invert the matrix in two
steps:
1
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1. Solve
Lx
i
=
e
i
and then
2. Solve
Uy
i
=
x
i
.
In this way the total cost for inversion would be the cost of factorization plus 2
n
3
. Unfor
tunately the factorization of a nonSPD matrix (with LU factorization) takes more effort
than that for a SPD matrix (with Cholesky factorization) which renders it inferior to matrix
inversion with Gaussian elimination.
But more on the importance of LU factorization in
coming sections.
In a way LU factorization attempts to follow in the steps of Cholesky, but recognizing that
it cannot take advantage of the properties of SPD matrices, i.e.
•
the matrix is not symmetric
•
diagonal elements can be zero or negative
For these reasons it needs some new concepts formalized.
1.1
Permutations
Because the upper left corner of the matrix may be zero it becomes necessary to change the
order of the rows. For example if we wish to start inverting
0
1
2
3
4
5
6
7
9
we will likely want
to permute rows 1 and 2.
We will denote such a permutation by premultiplying
A
by a
permutation matrix
P
, as follows
PA
=
0
1
0
1
0
0
0
0
1
0
1
2
3
4
5
6
7
9
=
3
4
5
0
1
2
6
7
8
Note that in order to express a row permutation as a permutation matrix, we have to adopt
a specific mindset.
Start with an identity matrix in mind, and then permute its rows to
accomplish the same permutation in the matrix it is multiplying. Thus we exchanged the
order of rows 1 and 2 of the identity matrix, because we wish to exchange the order of the
first two rows of A.
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 Spring '12
 Frances
 Cholesky Decomposition, Matrices, LU factorization

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