On the unique factorization of a non-singular matrix
by William M Lowney
A THESIS Submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree
of Master of Science in Applied Mathematics
Montana State University
© Copyright by William M Lowney (1954)
Abstract:
This paper describes a sequential method of factoring a square non-singular matrix into two triangular
matrices, The purpose of this factorization is to enable the computation of the inverse of the original
matrix to he more easily coded for machine computation, for the means of finding the inverse of a
triangular matrix is well known. Herein we deal with a matrix whose principal minor determinants do
not vanish/,- The method of attack is to make use of pivotal condensa-. tion of determinants and also
the use of extensions identities to form the sequential method of evaluation. The results ob-tained
indicate the possibility of this method being used for machine computation, OE THE HHIQ1
U E EACTORIZA T IOH OE A
HOE-SIHGUIAR MATRIX
■ ty
Jy
W I L I I M M 0' IOWEEY
A THESIS
SuUmitted to the Graduate Eaculty
in
partial fulfillment of the requirements
for the degree of
Master of Science in Applied Mathematics
at
Montana State College
Jj££adj Major Department
Chairman
Bozeman, Montana
June, 1954
L
-2*
4
^p.
table
of
contents
A B S T R A C T ...................... ..
3
INTRODUCTION .................... ..
4
FUNDAMENTAL THEOREM............. ..
5
SEQUENTIAL COMPUTATION . . . . . . . .
16
EXAMPLE.................................
20
CONCLUSIONS..............................
22
LITERATURE CONSULTED .................. ,
24
110363
'
ABBTRAGT
Tlais papej? deseriloes a sequential method o'f factoring a
sq%a^"@' iaoti^aixiguTa^ patsi^
triahgiilaa? Wat-ridW-o -TWd
■pWpdde Cf this factGldidatioW--id' :td ehahid the 'Cdmputatiosat h e 'itive'rse df the drigihai matrix: to he mere easily' e'eSM “f w
machine CCtopatatidB9, for the means of .finding t h e 'inverse of
a triangular matrix id well known» Herein we' deal with ~a .
matrix whose principal minor determinants.do not vanish. '
The method of attack is to make use of pivotal condensa*.
tion Of determinants and also the use of extensional identities
-to-, form ,the Seqnential method of' evaluation's' The results W *
t a i n e d ’indicate the- possihility of this method "being used for
m a c h i n e ..computation w
-
The P^Qhlem o f f i n d i n g the inverse of a matrix -is one of
utmost impowtaaee in ,pfa^tioaiigr all Wahshes .of applied math#*ematlsm*
'The- theOfatleal'solution Wein#' t M 'adjoint method is
almost wofthissS Sjfir aottihl .hWipEf'atiohi
G thW methWs ii' *$#,
task InOlnde partitioningapproximation^and statlstioal aetk*
bds*.- ^or p complete OmtTine Of ‘methods- m b - refer to Eors^the ts
article (Paige and Tanseky9 6}0 All such methods^ however^, ^
■fOOhire IahoriOn-S,Oo%pnt#ion and oonsidefahle- riaohin# tiaao* _
On the other Iiand5- if the matrix id- triangnlar5 ,the OOding for
machine OompntatSon has "been Set np and is W l l known.
-- ,
This paper treats of the factorIsation of & non^-singnlar
matrix into
conditions
> two triangular
■
. matrices*
e
! ', -ThS
. ^ 1.
.
- • . imposed
.i »
on the Original matrix; are.rslat-iwlgr 1ESah-?.. ,Wo dsrslop a -method of sequential computation of' the elements Cf the- trian­
gular matrices such that it may he coded for electronic com­
puters.
We- Shall show that the elements of Our triangular matrixoss are formed "by simple operations' on the two "by two detcrmi%
rants compiled in evaluating’the determinant of the matrix by
successive pivotal condensations^
Alw G
Then If A f. SG we have
^ and if B and G are triangular our inverse ,Is ,read!#
Iy Computed#.
-5-
THE E u h d a i -IEHTAL t h e o r e m
¥e take A = /a-ij) as our given non-singular matrix with the
added condition that none of the first principal minor deter­
minants of any order vanishes.
Ve shall factor A into two
triangular matrices, B and C, such that A = BC, where B and C
are of the form exhibited;
0
0
0
bjj bjj. 0
0
0
^3} O
0
b„ 0
"bji
(I)
I "k/i*."bnI "bflv
b nn
I
G/z Gli 0/y • . • • • Olfj
0
I
Czl Ca y • • • • • c-t/z
0
0
1
C1Y . . • e • C Jz1
0
0
0
0
(2)
6
Then
h„ C,j • e • • • • •
^ IZ
^iZ
C /i-
^0 U
f\z c Ztf
^iz
hj, Cjl^h3-L
^3Iiz
^3Ziz cUtHii
^Z C<
Om
• • • •
ji cZj^^3Ji Pti+^aj • • 1iz C"7+ '3Ji c-»k+ ^3Sj 9?»v
BC -
H i cIifH i c^ tHi * *
5T °n<" ^int^3Z
6V
However, if the two matrices are equal, we know that the
corresponding elements must he equal.
y
f
/ h /lr
h
“ KJ
That is
w3 ere p is the minimum of
(i.j),
for hy our method of construction it is obvious that
b ih = O
for K k , that Clfj = O for k> j , and C ic = I.
We shall prove the following theorem:
Theorems
(4)
I l« a # # « • a ir-z I^z a^tr
IaZZ
aU e •
•
• a K-Iz Ir-I
I
I^k
7
(5)
I it
#
•
ta^aMfrlfluf7)./
Jnll Srjl*
•
j?k
• a frk I
where we use the notation (Aitken, l);
a (< . . « . • • • , a,j a//f
a z/
IaV
*
•
•
•
•
•
•
♦
Jj a Zk
1Xl * * * a Jj a ii(l 1
aJ'
ajj &i*
a i(
a y a Lk
To begin our proof, we note that b„-a„ and, in fact, that
the first column of B equals the first column of A, thus b£,-atl.
IIow
a .j --
b„ C/j
c,j =
aJu'
•
au
The solution for b £1
is as follows*
a ix-
C12 + b£i
b,z = a £l - b £| cu =
- I^ i
J?j£x L
aU
^
a^x - a£, a.x
-8-
Solving for c 2j
Il
•
*1
a
C2j ’
cU +
a!j” a 2, H j J
a /i
I
aK
IJ111
a,,
J
Iai, Bill .
I a„ B 11I
To prove the theorem we shall use the first law of induc­
tion# noting that the column of V s
is to he computed first.
Assume the theorem true for b (j w h e r e
where j - I < k.
I ? j - I and C j
Then we have
h£
- Ia1, . . . .
|a„ • • • • • •
Cj.,
i
a/fj-,/
. . ..
i-i.a
Iall e . • • .'’j-.,,'-! a
I
*
I
Uow consider
(6)
Q
—
{a,. . . . . »a ;-1 , I a tj(
Ia„ . . . * • • a
I
.
B y use of our extensional identities (Aitkenf I) we get the
above equal to
9
K
• •
ia„
•
Ia 11 •
Ia ,, e e • a J-lJ 1 a j-',j/
eaJfyj-I /
• eaJ -V z a ^ »
Ia 11 •
« e8lJfyj-I I
/a.,
e a a J-Ij-JL a t j /
#
•
• »a Z-'y j-2 I
"Dividing the first row of the numerator by
and the second row by
|a „
. . ,Ayiii-J
Ia,, # .
we have:
0J-',)
1
t)£ j~(
lOfIi # # #
|a u .
k&ii, ,f t
Ia K • •
.
j-i
.B j-Iyj-I I
_
.9- lij J—.. "
• a /-iyj-i I
•
But this becomes, upon another application of the extensional
r,
I ii # # # ' I'l-j-S
i^j-2 J
I a ll • • • • Bj-J1J-Z. I
•
• ay-Jyj-J a JIyjl
I n • • •
/a,, .
0. L j ]
e
and following the same procedure as above this becomes:
*
Ia lt •
eaJ-IyJ-J./
O
C
•
•
o'
p.
•
I
Ia1,
3.0«*
I
0J-SJ
IfnI1 Si s s 9/i-3 ji-l cX/j I
I'it • • • aJ-JyJ-*I
ljSi-*-
IB h
e •
•
IaB
•
•
•
ai-j i-i a /
• aJ'SJ-J1
la„ • . 'aJ-Sj-JI
|a„
•
•
•aJ-SH
-
/
K-I
*
0J-fZj
"b ^zJ-K °J-'<,j
IaIZ • » 'a Z-IJ--V a J-SJ I
a Lj-jI
Ia1l • • 'a J-Sj-J I
Ia,,
• • •Bj-VfJ-V a Lj I
- I \ is r ~ J-KzJ
Ia II • * •a J-J-V I
c J-SJ
SJ-J
1Xj-,
Z-
-I
Iaii— s— s_«a I-VjJ-V a.j/
/a„ • • • av Sj-YI
la„ . . «aj-v.j-v &•ifI
IaIf • • • •
I
- Z
H SJ-K-
V/'fr CJ-Kj
tJ-AfyJ
A=,
or in general
IB-a # » «aj-ifj-i a ,J I
I8 It • • • • aJ-/,j-,1
I 'it > « »Rj-BjJ-H R/j I
Ib ii • • • • a
I
— y TDtK
/TM
c J-KzJ
11
If we let n = j - 2 we have
Q
—
/
-I Ii i/1)
/a„ a^i
la„ aJ
Cj'-k, j
Ar2z
Iall aXJi
Iall a l a « a t;/
Ia,, azll a„
I
0IJ
/a,, a ^1
-1
a„
la„ a,jI
Qftl
=
-
V
* Z-
A-=;
by -k 0J-",)
2
K'l
0J-^j
j-i
-
ati
Z _ h Vj'k
0J kyJ
- bt# 0U "
K
uJ-KyJ
Uow let j - k = I and we have
a LJ
-
2 1
b t-i
Cfj
"but this is a finite sum
—12—
Pa ij - Z
J
Jt-I
%
s
0 Ki
- i
k-i
*
Finally replace ^
by k
but this is the value of b tjil
for from our original matrices
a <-j = £7,
0Hj f N '
and so we see that
Q
c.
Ia // » , . - a , __ HujL
I3"!/ • • • • • aJ-Iyj-II
=
How we shall examine the C matrix.
Istif » .
Ia U • •
for
If we take
. a IjW lJ - /
Qiftcl
' a JOj-'
a JjI
and multiply by bjj , which we now know, we have
R — la,. » #
aJkjI
Jalt . • •ajWyJ-' av’jI
—
« faIi « . , a ,11
I8*/ • • • aJ-Iyj-If
I ,, . .
I6 !! • ' • '
Qjfl
•
I
Again using our identities, we have
la,, . . •aJH,I
IaII • • 'aZ-JyJ * aJ-IykI
Ia1, • • •OfijJ-I aJyj-II
la n • •
la(( . • 'aJ oJ-; I
.
» , R j - I jj. 2
a Jkl
13-
Ivlding the first row by
by
Ial, . • .
I
and the second row
(a,, . . .aHjJ , we have
I
c
which equals
(&U # » #Clj-T.i2.Q*IIrI
|a„ e • • •
I
la*i-S— S— '
(&„
Ia11 .
•
' a J - L iJ - !
•
•
•
I
J-W-Z RilrL
la„ • •
Jatt . . *a J-J7j ) aJ7Jjf
“
0 J-I7K
J-L1J4-L I
•
'a J-J1J-J
aJ-Z7ZrI
laH • • • -J-J7J-J a j> I
0 Z llM
Ia Il
•
•
* a Jl7J 11
Ia Il •
•
0Z-I1Jc »
' a J-J,/-! I
and following the same procedure we have
I
c/-V<
I' - if # # #c ^
i
I b u • • S • B j-^/-51
J//-2.
%_
Ia-I1 « #
ia n • • •
or in general,
I-JlL
a Jtc/
• aJ-J1/-j/
# #
la,, e e • • aJ'J-'I
bj,r*
0H k
-14-
)a,, # # #& *-/)^-/7 StipJ
lau • • • • aJ-P1J-*)!
jJ//-*
- f
k *
-I;/
If we let n - j - 2, we have
j-3
-Z
^j/j-Q- c>-*,k
/a,, a2x/
I--I
Iall H1J
)a„
aik/
/a Il a JxI
Ia Zf
a >/
bZJ--*
0Z-Z-*
1 =1
a i/
/u
A
v
-
I
I I#
/a«
a we/
a.i
aii«/
Z/Jg
Hr-*
cJ-AAr
0/-*,/<
1-1
j-i
aj k — a j I a <
a„
'
a ;K “
aA
-JI c'k ~
"
b AJ i
-
X
CH >
JL-i
bJjM
0Z-^zAr
cJ-A-
1-/
which hy a similar method as applied to the previous case re-
15
duces to
but from the matrix product
j
I a 11 *
« »a i/1
# i a i,
Iav • • •
or
R
Ip .. . . .a , /-I a
/S/i • * • . a jjl
*
«
I-'
0 ,1 i d
/Bi, • • • * ayj I
c
as desired.
Hence, if the theorem is true for the (j-l)th column of
B and the (j-l)th row of C, It vi3I be true for the jth column
of B and the jth row of C.
But we have seen it is true for
the first and second columns of B and the first and second
rows of C, and our induction is complete.
16
SEQU UTIAL COMPUTATION
Ve have now proved. It la possible to factor a non-singular
matrix whose first principal minor determinants do not vanish
into two triangular matrices.
Let us now consider the deter­
minant A from the aspect of pivotal condensation (Aitken, l).
a i/7
a,? • '
B 1T
(7)
/Al
-
ax,i°
S
• • # e $ #
#
where
#
a „n
•
B y pivotal condensation we have
P-u
a^|
I IalT
Pu
(a)
H-.
J_
(a,rrL
Iaw0
aU I
•
S • /aC
a W I• # e |a,°
aJ/
a WiI
•
#
#
s
# # e
#
•
#
e
#
• • e #
•
Bpil
Pu
aAij • # * Ia,?
|a,r
Let
(9)
-
K
then (8) takes form
<
a,ie •
aVl aj.
*
* S/H
, • •
I
•
V *
* • • 3j F t - ' , n - t
a ,;/
iJ e
17
Hence, we see that
(10)
.
a*k
••••
•
• e#
a#
*#a #-**
•*,
,•
1 • • • It-KjM-U
M -- JLW
a*
a,* . . .
•
s
s
e
where
ISi,
k-f „ k*',
t
I
(H)
k^l
a,7-1
S
• r
In this notation we shall see that B takes the form
a,7 0
aj
a,r
/
aJ, a*
a„°
•
•
V
2j l u
a,,1
Osee . . 0
0 .. .
a^e . .
a,,'
# # * # •
a/I-I1X1e e •
•
•
Q
and that C takes the form
I
0
C
0
•
S
•
0
0
0
17'/
«'*•
/I
• S l U-
I
18-
Hence, we see that
j-i
(12)
"b U
^
S,L-i
,I
a,rz
t"-i
(13)
cij =
a- ^ ; ^ 7
^
Il
To prove the latter true we refer to (ll)
a
k
ij
k-« I
a ;>/. I<-t I
7-x
a
and following the extensional identities of Aitken we have
k-i I
a w/> I' I Ia K • • * Kkl la« • • •ak-Z,k-1 a i+Ky3+kl |
- |a1( .
hut
(14)
•
*a IHjK-I I
K I * • eaKk
a,V =
Iali * . •n Ir-Iy
k
a ij '
/a /, • • •aKk a LtkyJtkI
• • •a Jrfc a CtkykfI I
a if •= Ki
a CtkyJtk)
m
O
Iar
#
How from (4)
l0Ukftl
--
I B 11 . . •a a:K a Lt-Ic^k I-/1
)a„ . • •a Kk/
K
= -a JJa Il
or the same form as
(12).
— 19—
Prom 04) we see that
a Ij
K 1 • • * Kk v I f A j J t * )
~
hut from (5)
^ .
l,
..i.
^
I. #
#
#
/r Ar & /c»/ .
IaIf » • • • 8-
a
'
or (13) as desired.
a,A
Hence, we see that the elements of B and C ma y he compu­
ted sequentially as a product of 2*2 determinants hy computing
the Jth column of B and then the Jth row of C.
We see that the
kth column of B is found hy dividing the first column of the
(k-l)th pivotal condensation of A hy the first prlnci al minor
of the (k-2)th pivotal condensation.
Likewise, the kth row of
C is found hy dividing the first row of the (k-l)th pivotal
condensation hy its first principal minor.
20-
AH rXAMFXE
Consider the matrix
A S
2
3
1 0
1
2
0
0
1
3
0
1
1
2
1
1 2
1
2
1
How we have* hy pivotal condensation
1 - 1 4
|A|=b
1
2
2
2
4
9
-I
4
-I
1
1
2
1
21
2
O
O
O
O
I
3
2
I
2
O
I
2
I
I
O
O
O
O
I
-I
4
I
2
BC ?
O
I
2
O
O
O
O
I
-3
I
2
3
-9
2
-5
25
2
O
O
O
O
I
13
25
I
-I
2
O
3 -M
25
O
O
O
O
I
G0Q8&B8IOBB
Xn this paperg the respite 'derived, show a simple sequent
tial method of factorization of a non»singiular matrix whose
'' -
first principal minors do not vanish=.
'
i
.
■■ ■
A method for computing
the 'IpverSes .of the resulting triangular matrices Iihs.."been .
oodea for machine computation (##a&er* Punean and collar*
10881* '
:
Por computational purposes j, We note that we are evaluate
ihg two. by two determinants to solve for the jth column Of B i51
and using these results to solve for the jth row of 8* wherein
again we have only to evaluate two by two- determinants*
Ihe
significant advantage Of this method*, the# is the simple ee*
quentlal method of attack=.
We also, note that by weakening the conditions of our original matrix further work can be done=
BOf example* change
ing the non-~vanishing condition of the first principal minors
would lead u&* in effect* to results obtained from interchange
Ing rows by elementary operations*
1
•
»
The possibility Of computing the inverses of the triangu-*
V
Iar matrices in a similar sequential method could lead to an*
other line of research*
This method may have an advantage in
enabling the computation of A inverse directly through proper
codification*
The next, condition to be eliminated could be the non*
«*23**
Singularity of our original matr!%*
T h i s method probably
could he applied to singular matrices.- ' In this case it would
seem that the B matrix would have n-r hull column vectors*
where r denotes the rank of the matrix.
*8#
LITlRAfTO CONSULTED
it
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2.43 p^a j,:,lnterseieiice Publishers fada g B w Y o r k b:, ■
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PrdOti A m e r 0 Mathb Soc0; £§288*BQO0
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,., ;- S37 p p b # A d d i s o h ^ e e l e y Press IntiUV Cambrlige^ Massy"8*
Petrief
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MODERB ALGEBRA ABD
371 ppb* Chelsea Publishing Cd06 Bew 'Ydfk*
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MONTANA STATF im r v c B c , ™ _________
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Lowney, William M
On the unique factorization oi
a non-singular matrix.
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