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ELA
FULL RANK FACTORIZATION AND THE FLANDERS THEOREM
∗
RAFAEL CANT
´
O
†
, BEATRIZ RICARTE
†
,
AND
ANA M. URBANO
†
Abstract.
In this paper, a method is given that obtains a full rank factorization of a rectangular
matrix. It is studied when a matrix has a full rank factorization in echelon form. If this factorization
exists, it is proven to be unique. Applying the full rank factorization in echelon form the Flanders
theorem and its converse in a particular case are proven.
Key words.
Echelon form of a matrix, LU factorization, Full rank factorization, Flanders
theorem.
AMS subject classifcations.
15A15, 15A23.
1. Introduction.
Triangular factorizations of matrices play an important role
in solving linear systems. It is known that the
LDU
factorization is unique for square
nonsingular matrices and for full row rank rectangular matrices. In any other case,
the
LDU
factorization is not unique and the orders of
L
,
D
and
U
are greater than
the rank of the initial matrix.
We focus our attention on matrices
A
∈
R
n
×
m
with rank(
A
)=
r
≤
min
{
n,m
}
,
where the
LDU
factorization of
A
is not unique. For this kind of matrices, it is
useful to consider the
full rank factorization of
A
, that is, a decomposition in the
form
A
=
FG
with
F
∈
R
n
×
r
,
G
∈
R
r
×
m
and rank(
F
)=rank(
G
r
. The full rank
factorization of any nonzero matrix is not unique. In addition, if
A
=
is a full
rank factorization of
A
, then any other full rank factorization can be written in the
form
A
=(
FM
−
1
)(
MG
), where
M
∈
R
r
×
r
is a nonsingular matrix.
If the full rank factorization of
A
is given by
A
=
LDU
,where
L
∈
R
n
×
r
is in
lower echelon form,
D
= diag(
d
1
,d
2
,...,d
r
) is nonsingular and
U
∈
R
r
×
m
is in upper
echelon form, then this factorization is called a
full rank factorization in echelon form
of
A
.
In this paper we give a method to obtain a full rank factorization of a rectangular
matrix and we study when this decomposition can be in echelon form. Moreover, if
the factorization in echelon form exists, we prove that it is unique. Finally, applying
∗
Received by the editors January 21, 2009. Accepted for publication July 9, 2009. Handling
Editor: Joao Filipe Queiro.
†
Institut de Matem`
atica Multidisciplinar, Universidad Polit´
ecnica de Valencia, 46071 Valencia,
Spain ([email protected], [email protected], [email protected]). Supported by the Spanish
DGI grant MTM200764477 and by the UPV under its research program.
352
Electronic Journal of Linear Algebra
ISSN 10813810
A publication of the International Linear Algebra Society
Volume 18, pp. 352363, July 2009
http://math.technion.ac.il/iic/ela
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Full Rank Factorization and the Flanders Theorem
353
the full rank factorization in echelon form, we give a simple proof of the Flanders
theorem [4] for matrices
A
∈
R
n
×
r
and
B
∈
R
r
×
n
with rank(
A
)=rank(
B
)=
r
,as
well as of the converse result.
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This note was uploaded on 02/22/2012 for the course EE 441 taught by Professor Neely during the Spring '08 term at USC.
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