UNIVERSITY OF NCRTH CAROLINA
Chapel Hill, N. C.
JOINT REPORT
Contractor:
B. G. Greenberg
Contractor:
Department of Biostatistics
Project Number;
Department of Army Project
No. 5B99-01-004
Ordnance Rand D Project No,
TB2-0001
S. N. Roy
Department of Statistics
Mathematical Sciences Directorate
Ail' Force Office of Scientific
Research
Washington, D. C.
OOR Project No. 1597
Title of Project: "Estimation of Pa~ameters
by Order Statistics"
Contract No:
Contract No:
DA- 36- 034-ORD- 2184
AF 49(638)-213
AFOSR Report No.
Technical Report No. 10
December 22, 1959
Evaluation of Determinants, Characteristic
EquatiODS aod Their Roots fot' a Class of
Patterned Matrices
by
S. N. Roy, B. G. Greenberg and A. E. Sarhan
....
'
..•
. EVALUATION OF DETERNINANTS, CHARACTERISTIC EQUATIONS
AND THEIR ROOTS FOIt A CLASS OF PATTERl\lED NATRIGES
by
SoN. Roy, B.G. Greenberg and A.E. Sarhan
University of North Carolina
introduction.
In previous papers
!J.J, [)J ,i>J,
the
authors have examined matrices w:1.ti:J. special but common patterns and
noticed that they were amenable to simple and
sion.
ra~id
methods for inver-
The present effort is concerned with the evaluation of determi-
nants,characteristic equations, and characteristic roots for the same
class of specially structured matrices. Some of these operations upon
•
the matrix are not only mathematically related to the process of 1nversion, but are often required, in addition to the inversion process
in such fields as analysis of variance, response-surface fitting and
multivariate analysis, and on data under various types of models,
"normal" and "non-normal.. It
It turns out that determinants of this class of patterned
matrices can be evaluated ;uore readily than is apparent at first
glance.
A by-product of this process is to demonstrate methods of
reducing patterned matrices, such as by triangulation, to facilitate
rapid evaluation of the determinants.
It also turns out that for this
entire class of patterned matrices considered here, the characteristic
equations can be obtained more readily than apparent at first glance.
Indeed, for many of these patterned matrices the characteristic equations can even be written down at sight.
In some but not all of the
cases considered here the latent roots are obtained as readily.
•
.>-
2
Unfor'tunately, aocording to our present knowledge, the latent reots
in ·the more difficult cases appear to be determinable only
by~he
numerioal solution of the oorresponding oharaoteristic equations.
However, even this would be an improvement over the methods of
numerioal evaluation of latent roots given by suoh workers as
Aitken [1J and Hotelling LV, if those elegant methods, intended
for general matrices, were applied to the patterned matrioes oonsidered
here.
Ivlethod o
The methods used for evaluating determinants are
elementary and well-known. For example, by subtraoting linear functions
of oolumns from one another, the struoture of the determinant can
be modified and reduoed to the triangular form.
This allows im-
mediate evaluation sinoe the determinant of a triangular matrix is
the product of the diagonal elements alone.
Another devioe for reduoing a matrix to a more oonvenient form
is that of
partitionll~.
It is known that if
p
A ...
R
then
A·
3.
-I
I 81
I
t •
p .. Q S-1 R
Evaluation of Determinants.
(3.1) Consider
th6·~ants
Da + a £. E' where Da is a diagonal matrix with ai 8S diagonal
1
i
elements (i=I,2, ...n) and £. represents. a vector bI b2•••bn 0
A
r
In its more conventional form, the determinant appears as
fj -
,\+CIbI 2
CIbl b2
CIb b
l 3
• • •
ab 2b
l
a +ab 2
2 2
CIb2b3
• •
ab b
31
CIb b2
3
a +ab
3 3
••
n
ab b
3., n
• • •
•
•
CIo• b
n 2
CIbnbl
CIb 2b
•
II
0
•
2
CIblbn
•
a~nb3
.a ~CIb 2
••
•
n
n
0
Subtracting from the jth column, the product of the first
b
.
column times ~ (where j>2), one obtains
1
-
~+CIbl
A-
2
0
•
~
0
-a b 2
l
b
- )b3
b
1
l
G
. -a1bn
on
1
0
•
o ••••••• o ••• o ••
CIb2b
l
CIb b
3 I
••
•
CIbnb
<)
(l
0
2
•••••••••••••••••• ob.
0
• •
•
0
..
•
0
41
0
e-
•..
•
O
•
(I
.
0
0
0
l
a
"
~
••
•
0
a
3
•
0
•
• <)
0
• •
0
•
•
•a
n
o
....
b 2
b 2
3
+ CIa...L(a2- + ...:.....+•
a
2
3
0
4
b 2
1 . 3 • •• an )(.1 + a(-!a1
= (a aaa
=
(
1 + a
+..!..
+•• ~+
a
2
2
n b. ) n
Z a~
1T ai
i-1
Example:
b 2
b 2
an
~
n ')
•
i=l
1.
In the multinomial distribution, the determinant
of the varianoe-oovarianoe matrix would appear in a form as follows:
A ., Pl (1-P1)
-P1P2
-P1P3
-P2Pl
P2(1...P2)
-P2P3
-P3P1
wop P
3 2
P (l-P ). • ,-P Pk
3
3
3
0
•••
I)
•e-
•
•
-P~l
-P~3
-PIt2
•
(I
e
• -P1Pk
• .-P2Pk
,•
• )
Pk (l-P
k
•
I
In this oase" a
i
I Pi
A .... D
represents
~,
- E E,J
I
J
= Pi"
(
= 1 -
n
= k"
k
Z
a • -1 and bi .. Pi' Thus,
p)
•
i-1 i
or a unit vector, then
~
!oJ = J" where J .. a(nxn)
matrix with all elements unity. Thus,
Example. Oonsider the determinant of the matrix which ocours
in the anlysis of variance for the one-way classification with un-
equal frequencies in the different classes •
/).
""
-
-1~
-
...L+..!..
-
-nk
-
....L+
~
.1
1
ri
1
~
,n
k
k
~
1
~
'1
-n1+ .......
nk
3
•
•
nk
•
"
•
1
•
~
•
1
•
1
-
-~
-
~
~
l)
• •
~
1
1
0
•
~
•
~
-1~
•
-1~
0
.Jn
k
•
~
•
<I
•
II
0
•
J..., + .l:...
-rr;
,n
k
If in (J.,l), &i "" A, for all i, then
a1 + abb l
--
A•
-
J ""
n
2
an (1 + 2. ' ! b. )
a 1-1 J.
n
an- l (a + a Z' b 2)
•
i-I 1
By
(O<dS~(J.l.l)
and (J.l.2), one obtains a form of
matrix very frequently encountered in response-surface fitting as
well as in the analysis of variance. That is, ai -: 0.., bi I::. • /
n l
aI + a.J ( "" a - (a + an)
•
!,
than
6
Example: The determinant of the matrix occurring in the analys:J.s of variance for the one..way classification with equal frequencies in the different classes iss
A ..
:3
1
1
1
:3
1
1
1
3
•
.•
•
•
1
Thus, a
•
1
• ••
• •
1
• • •
a = 1"
n•
=
121
+
JI
•
•
•
•
1
1
1
0
o ••
•
•
•
=2.
•
•
o
3
n" and
A • 2n- l (2 + n)~
(3.2) Determinants for the diagonal matrix of type 2&
The diagonal matrix of type 2 has been defined elsewhere
..
......
,.. t . '
.
i:ts detaft't1iJ.nant 1.· aslol:t.ot$l
.
d . •• k
i
A .. a b c
'~"."
._.~
b
~b)
\1.e
l1d
• •
c
lJ;c
ac
ad
• ...
ak
fit
atl
p;d
ad
vel
• ••
yk
r(
d
k
•
•
•
iik
~k
yk
f/
iii
af/
rK
•
0
•
•
••
•
~
.. .
0
•
· • • nk
Multiplying the first column by
n/.
.~
~2-7,.and
•
itt
8'
•
and subtracting the result
from the second column, multiplying the first column by a and subtracting this result from the third column, and so on, one obtains,
7
e
Ii ==
a
b-al-l.
b
0
:1bZC-ba
0
0
0
d
0
0
••
k
t
V..,ao
\TrId-by • • • tMk-bn
~d-cr • • o a.k-cn
_{(...bO
yk-dn
yV-do
0
•
• •
It
9
c
,
at-oo
•••
III
••
••
0
0
0
• 0•..
0
0
0
0
0
an-k
ao-I
a
•0
c
k-an
d-ay •
c-sa
•
•
(>
nj-ko
•
aa"'c
ay-d
0
ba.-I-I.0
bY-l-I.d •
bn...1J.k
bO-l-I.i
b
0
0
cr-ad
••
• ••
on-o:k
oo-at
°
e
0
0
0
• • •
dn-yk
do-yi
d
•
i
0
0
0
•
•
0
0
0
BfJ.--b
Ii ==
•
•
•
•
Ii ..
(~
•
~
•
0
0
r.,
0-
•
Q
~
•
•
•
• •
- b)(ba ... I-I.0)(oy - ad)
•
CI
•
•
0
•
~
Q
0
k~nl
k
0
0
l
(ko - nt)
f
•
It may be noted that this determinant is the produot of a series
of (2x2) determinants obtained in the following way:
Put on one line
the elements of the first raw (viz o a, b, 0, ••• k, [); on the second
line immediately below, plaoe the coeffioients that make thj.s a type
2 diagonal matrix (viz, 1-1., a, y, ••• n, 6); make the (n+l)th element
equivalent to zero, and the (n+l)th coeffioient equal to unity. Thus,
one obtains
a
1
b
°a
d
.0.
k
•••
n
o
8
1
8
and
A • (~.'b)(ba - ~o)(cY - da) ••• (k6 -
nl;(l)
0
It oan be seen that the overall determinant will be zero
if anyone of the individual (2x2) determinants is zero.
words" the matrix is singular if
In other
b" ba • ).I.c" etc.
S(J. •
Example. Consider the value of the determinant resulting from
the
~ajor
part of the variance
ma~rix
of the order statistios in s
SeIPle from the rectangular population.
A-
n
n-l
(n-2)
n..l
2(n-l)
2(n-2)
•
n-2
2(n-2)
3(n-2)
,
•
•
• •
•
•
•
•
•
•
1
2
3
•
In this oase,
••
2
.•
•
~
b • n'l"l
a • 3
c • n-2
Y-4
••
t·
3
•
•
•
•
n
.. 2
t,an
1
Rewriting in the form of
L2n ..
1
s-n
d • n-3
A•
• • •
n
1
n-l
2
b-2
3
n-3 • • • 1
4 • • • n
J•..
(n-l~
L3(n-l) .. 2(n..2E [4(n-2) .. 3(n-3)
•
(n+l) (n+l) (n+l) • • • (n+l) 1
_
(n+l)n-l
•
0
1
1
9
The determinant of the type 2 matrix given in (3<>2)
may be rewritten in another, more oonvenient form, as
" "
~
.
r
°201~
°2 :,al·~)·' °2 03(al +a2 ) • •
0 a
°3 1 l
°3 02(al +a2 )
•
•
2
°3 (al +a2+a )
3
•
• •
•
•
et
° nl
o·~ °n02(~+a2)
°20n(~+a2)
.03On(~+a2+a3)
0
•
••
cn03(al+a2+a3)·
It
•
cOn2 (~+a2+a3+.
• .+an )
The matrix related to this determinant was shown in [$..7 to
be
expre~ble as
where T is a lower triangular matrix with all non-zero elements equal
1
to unity and D and D are diagonal matrioes as defined before. Eaoh
0i
8i
of the five matrices has a deierminant equal to the produot of its
n
2
diagonal elements, so that I:!" iT a 0i • The value of this deteri=l i
minant is zero, of course I when any a or 01 is equal to zero.
i
This determinant oould have been evaluated from the prQvious
result given in (3.2).
are as follows:
The transformations making this possible
0
10
•
•
~
8
on a +a2+a +o
( 1
3_
°1
(303)
o+a
n)
•
&11 _ _
~
Consider the determinant
aJ
mIp
I
where It • identity matrix of rank
t~
Using the partitioning principle in Seotion 2, this mar be
written as
I
A = mIp 1/ kIn"
i J(nxp)(~ )J(pIA) 1
2
A
&11
niP { kIn ... ~iiP.·
J (nxn) (
The results from (3.1¢3) can be used
toe~ate
directly
2
the latter part of the right-hand side by letting a
&11
k and A = .amP •
Therefore,
D
n 1
A • nr k -
a2
(k - !Je
m n) , or
2)
A-nr0-1 k n-l (mk-anp.
0.4) Consider the determinant of n rows and columns of the form
such that
11
A• J g
-
be
Then,
A-g
b-g
f
Again, the results of (3.103) oan be used where a = o-d,
2
and A • dg-b • Therefore
g
d
g(c_d)n-2 [(C-el) +
G
(~f.;b2) (n.-l~
2
b. ... (e_d)n...2 [gee-d) + (dg""b ) (n..l)]
Example 0
•
A typical matrix from a problem in response...surfaee
estimaticn [2.1 may have
A ..
a
determinant
as
follows:
27
34
34
34
34
~
2
4
113
~
34
~
"2
T
34
J¥
T
113
161
"1:
27
=-
161
113
T
-
34e l
34!
Here,
g ::
27
b =
34
n=4
o
12
Therefore
A Special case. If the determinant considered in Section
(3,,4.) is simplified by making b - d and g - 0, we have
~~) ~-l
g
A ..
be
+ bJ
I
A.
2
(g... b)n'2[ g{g..b) + (bg_b )(n-l) }
A ==
(g..-b)n,.,ll g + b{n-l)
1~
Example 1. Consider th$ '.ir...:'inite matrix used 1.."1 Section
where the determinant was found to be 2n-l(2+n)
<'~ol(3)
0
This same determinant may be rewritten in the form,
A
•
3
-e
et
21n-J. + J
where g • 3 and b -1 "
Then, A = 2n-l(2+n). This
CaB
be generalized by sUbstituting
(gwb) for s, and b.for A, in the results from Section (3 Ql.3).
Example 211 The determinant l. coftlJ;1der&d. .U'l.. $e:etibl'l .(3·.4) .~~ ~on1y
s portion of the overall determinant from the particular problem considered o The original deterrr.inant was)
•
(30)
The matrix whioh oocurs in the analysis of varianoe of
a latin square experiment presents a determinant of another inter...
esting form as
6 •
II -/
mIk
J
J
J
mI _
J
J
J
k 1
mI
k-1'1
r
~Il mlic-J.
J - (~\) [J J]
,,\j..JJ
jJ
k
Ii •
m
l- ~~~
L~k_l J
IlU1t-1
~J
kJ
kJ
k
(1 • -) J
•
k
(1 • -)
m J
k
-
Ii· m / mIk_1 +
I f(m~.l~)
.
k
(m~_l • iii J)
-
(1~)2J(mIk.l+)·lJ
.
. ..
.
-.. ." . .
-,
By using the results of Roy and Sarhan [lU, the matrix (mI _ ~
~~',
",
I
~
~ J)
k l
oan be inverted and the first determinant evaluated by the use of
Section (3.1.3) so that the result is
•
1$
(!)(k-l~(DiIk-i~)-(m;k)2J~~Ik"'1+'
k k 2
A = (m )(m .. ») m ".
m~.
t
Using the results from Section
A•
m3
1
(3~1.3)
l ;? [lal
2
k 5
- ('?k +k) m -
m(m
k
J] J )
-k +k)
2 2
again, this reduces to:
:tt)1J
+ (m-k)2 (k-J.>(;+
t
2
= .,3k-5 (m - k+1) m + meW) - 2k(k-J.)
1
Consider as a special case of (3.$) the determinant given by letting
m .. k, viz.
J
A. J
J
J
J
The result is clearly seen to be
(4)
Characteristic Equations and Roots
The characteristic equation of a matrix A is defined by
1 A .. Unl" 09
The solution of this characteristic equation gives
the charaoteristic roots which are represented by
Consider the matrix, M,
~, X2'~~3'o
•
O)A
n
0
•
J.6
i...
(4.1)
2
M:: a1+o:b1
o:b b2
l
O:blb
I
O:b b
a +o:b 2
ctb b
2 J
ctbJbl
ctb b
Jiii 2
aJ+abJ
•
•
ctbnb l
•
•
2 l
2
II
M=
D
ai
2
•
+ o:b b l
2
O:bnb
o:blbn
• • •
O:b b
2 n
.
O:bJbn
e
•
•
•
O:b b2
n
•
•
-/}
J
J
0
•
..
•
.•
•
•
o
a +ctb
n
2'
!
n .-I
j
- - .. :
The characteristic equation is, therefore, given by
\ D(ai-X) + 0:'£
£1 I• 0
..
By use of (J.l), this leads to the oharacteristic equation
b. 2
n·
n
-:1+0:(2:
~ )fJr(a.-X)=O
(
.
. 1 a~.-X)i
1
J.
1= . J.
. =
Example:
The characteristio equation for the example given in (J.l) is
i
k P 2 ,k
} 1 .. 0: 2: :!--l ri"·{p.-X) :: 0
I
X.)IJ.=010
., 1
\. i I
= p.-,
J.
0
As a special case of the matrix given in
vector.
(401), let b
The characteristic equation th~n beoomes
,
n
1
) n
X ) ~ IT (a. -1\) .. 0
. 1 a...
J.
1=
J.
.J.J.a1
}t 1 + 0: ( Z
If all a a a, (4 Q lcl) reduoes to
i
0
&
a unit
•
17
Solution of
(4~l.3)
yields (n-l) characteristic roots all
equal to a, and the n~ root is obtained by equating the quantity in
Suppose that only
~
n
~
the brackets to zero. Thus,
- a + a Eb
n
. 1 i
J.-
2
g
• a2 (-a, let us say.)
In this case, a
is one of the characteristio roots and the equation can be reduced
one degree to solve for the remaining (n-l) roots.
Suppose that a =a -a (= a, let us say.)
l 2 3
In this case, a
is a repeated characteristic root and the equation can be reduced two
degrees to solve for the remaining (n-2) roots.
By induction, i f al =a 2,.
follows.
0
ol=an' and the result in (4.1.3)
£ = a unit vector, then (4.1.3) reduces to
If all ai-a, and
t,(- a.,.).)
-.
+ an J (a-A)
n-l
• 0
ThUS, there are (n-l) characteristic roots equal to a, and
the
n.!: root
is (a+an)o
Example t
M=
The charaoteristic roots of the matriX
-- 5
:3
:3
L3
3
5
3
:3
:3
3
5
3
3--
3
3
[21 + :3J
4
I
5--
can be found by letting a-2, a 3, n=4.
D
-
The four roots are therefore
•
•
18
Since the sum of the characteristic roots (viz.
2+2+2+14-20) must equal the sum of the diagonal elements (or trace),
this is obviously confirmed.
Furthermore, the product of the charac-
teristic roo·ts (viz 0 2x2x2xJ.4=1l2) must equal the determinant, and
this is also confirmed o
(4.2)
Consider the diagonal matrix of type
(302 l)0'
Q
~
occurring in Section
In this case, the characteristic equation can be expressed
as
where the usual definition is used for the traces of different orders*
Thus,
and
n
2
trA-IA\=l'To a.
i
n
!
i=l
1
Q
To evaluate the traces of ciher orders, the matrix is first
rewritten in what was previously shown to be an equivalent form, viz •
•
To obtain from this, for example, tr A, it can be observed that
2
tr2A is the sum of the seoond order principal minors of this matrix.
A typical second order principal minor formed by the rows i and j and
columns i and j (i<j) of the original matrix is given by
19
•
• • •
..
Va.1
o • • •
000 • •
•• va:-J
..
0 0
•
&
0-·
·°11 y~-
.0...1' 'Ia 2
va
l
va;
0
/I
"
0
•
va.
1
0
•
va
i
vai +
~
0
•
•
•
•
0
\fa.J
0
0
0
0
•
0
•
•
•
0
0
/I
This, in turn, is equivalent to
Thus,
In general, one can prooeed to find
n
i ..1 i 2-1
3
2
trtA = ~ ••• ~
E 0 2 o. 0
12
it""t
i =2 i -1 i l
2
1
, (a.l
.I.
1
+1+6:·"
••
L
.+a.
;1.2
I
•
)(ai +1+· • ,+si' ). •
2
3·
where
(~<~< • • • <i
t)
For illustration, the case when n=4 may be considered by
•
.
•
20
indicating the characteristic equation.
It is as follows:
l
4 3f 2
2
)
2
)
2
)('
A -A c1 a1 +°2 (a1 +a2 +°3 (ai)a2+a +°4 (a1+a2+a3+a4 )
3
21 2 2
2 2
2 2
+A I cl 02 a a2+cl 03 ~ ( a +a )+:c 04 a1 (a2+a +a )
l
2
1
3
3 4
(
2 2
.22
c2 03 (~+a2)a3 + °2 °4 (~+a2)(a3+a4)
2 2
+°3 °4 (a1 +a2+a3
122 2
..,Al01 c 2 ° 3 ~a2a3 +
(
"J.
)t\j
222
°2 °4 ~a2(a3+ak)
2 2 2()
2 2 2 (
\~ (
+02 03 °4 a1 +a2 8 3a4 + °1 c3 c4 ~ a2+~/"4 j
2 2 2 2
+01 c 2 03 04 al a2a3a4 • 0
Example:
.;)
Consider the matrix of diagonal type 2 given by
.l
2
6
4
6
o
The matrix Y. represents the variance matrix of order statistics
for a sample of size 4 from the rectangular distributione
1his form of
the matrix is the same as that in Section 0.2 0 1) by putting
01 •
4
~ .. 1/4
02 .. 3
a .. ,/12
c .. 2
3
04 -1
a3 • 5/6
2
a ..
4
,/2
•
21
The characteristic equation for this matrix is
For a general n, if all cils are equal to c
and it all a
i
I
s are equal to a, then
1 2-1
-1:
1 (i2..,i )( i -12 ) It
1
1 3
i =2 1 ...1
i -1
o
0
3
E
2
•
a(
1 -i ... )
t t 1
1
Thus, for n=4, tho characteristic equation reduces to
If only the oils are all equal to c, and a remains as such"
i
the oase for n-4 will have a characteristic equation as follows:
For example" consider the portion of the variance matrix of order
statistios for a sample of size
A , - J./J.6
I
,
i(synnnetric)
I-
4
from the exponential distributiono
1/16
1/16
1/16+1/9
1/16+1/9
1/16
1/16+1/9
1/16+"J./9+1/l..J, 1/16+1/9+1/4
1/16+1/9+1/4+1
This has a pattern typical of the type 2 matrioes and can be
derived by equating
Let
0
•
22
°1· 02'
= '0J = c4~ '= 1/16,
8
1
1111
2- 119, a3 1/4, a4=1 .
The characteristic equation is clearly
2
>..4 .. 25/.1.2 >..3 + 115/1L4 >.. .. ll/144 >.. + 1/576 - 0
0
Alternatively, if the ai's are all equal to a but the 0i's remain
unaltered, the charaoteristic equation for n=4 is
r:
II
A4 .. >..3a o12+2c22+3 c32+4c42J'- +A2a21c 202 2 + 2cl 2°3 2
L
J
,.)
2 2 2 2 2 2 2 2
3- 2 2 2
+3 cl 04 +2°2 c3 +4c2 04 +303 04 .. >..a 1.°1 °2 °3
2 2 2
2 2 2
2 2 2-/
4
2 2 2 2
+201 02 c4 +202 03 c4 +201 03 04 _ + a 01 02 c3 c4
(4t1.3)
1#
0
Characteristic Equations and Roots of Silnply Par.titi.oned Matrices.
Consider the matrix A,
(40301)
aJ
mI'p.-
•
The determinant of matrix A was shown in Section (3.3) to
2
n1
be equal to k .. nf""'1(mk .. a np)"
By replaoiag· k by k-A and m by
~>..,
the characteristic
equation.is seen to be
Therefore, there are (n-l) roots equal to k, and there are (p-l)
•
2.3
,.
ro?ts equal to m" and the remaining two roots are obtainable from
2
solution of (m-A)(k-A) ... a np • 0 , being thus equal to
- - - - 2 ----2 (k ... m) :. Y(k ... m) + 4a np
o
•
2
(4;:!~)
.Q.haracteristic Equations and Roots of Other Partitioned Matrices.
Consider the matrix A,
-=
A • \ a
b e
Where e
-
unit vector.
In Section (3.,4), the determinant was shown to be equal to
By replacing a by a *"A and c by Co-A, the characteristic
equation is
or
There are (n-2) roots
equ~to
(c-d) and the other two roots
are obtainable by setting
(c-d-A)(a-A)
Example:
+[ d(a_A).b2 ] (n...l)
=0
0
The characteristio roots of the matrix given in Section
(3.4) can be found as follows I
Let A=[2 7
34
.34 e
~
t
]
209~ 13 + 113/4 J _
•
•
24
Then, the charaoteristic equation ist
(209/4
~ >.i ~ (209/4 ~ >')(27 -
>') + 3 [ 113/4(27->') - o4l]
r
Two roots are equal to 209/4" and the other two oome from solution of
As a speoial case, let
A
-l-a
and a=o.
b~d
Then
-
b e
Working with the determinant of this matrix and replaoing
~
by
a-A, or by going direotly to the characteristic equation given
earlier and sUbstituting for
0
and d, the result is
o
ThUS, (n~l) roots are equal to (a-b) and the ~ root is equal
to ~+b(n-l)l •
Consider the matrix from (39401) where
Example:
:3
1
1
1
A- I
:3
1
1
1
:3
1
1,
1
1
1
3...1
-
f3
I
-at
j
I
I e
I
L-
2I +
3
1
Jj
so that a=3, b=l, and n-4. Three roots are equal to 2 ~ and the fourth
root is 60
:Example:
Consider the matrix given in Section (J.4 Q l)o
a
•
•
-25
....
-"
A
.1
I
I
34
34
",
I
34
I
113/4
113/4
I
I
113/4
1
-
....-.-! -
--
_."
21
34
34
161/2
113/4 113/4
34
113/4
161/2 113/4
34
113/4
113/4 161/2
34
34
or
0-
T
•
Therefore"
Six of the characteristic roots are given by the Pi'S,
vi~.
three roots equal to 34 and three equal to 113/40 ~e remaining four
roots are those obtained previously in the first example of this Section.
(4Q5) Consider the matrix of Section 3.5.
Let A =
r
J
J
J1lI _
J
nllk
l:
k 1
J
mI _
k 1 .J •
The determinant was given as
3k-5 (m-k+l)t m2
m
f
+ m(k-1) - 2k(k-l)
Jl
•
•
•
26
By replacing m by
m-~
the characteristic equation is
There are (3k-5) repeated roots equal to m, one root equal to
(m-k+lt, and the remaining two roots obtained from solving
2
(m-"-) +
or
(m-~)(k-l)
- 2k(k-l)
llS
0
...
~ . ' (2m+k-l) + V-tk-l){9k-lJ
2
If the special case is considered whereby m=k, there are
Ok,,,,) repeated roots equal to k, one root of unity, and two
roots equal to
~=
(3k-l) ; V(k-l)(9k-l)
2
o
•
27
References
A. C. Aitken, "Studies in practiCal mathematics."
of
£2J
~oX.
Proceedings
Soc. of Edinburgh, Vol 57 (1937), pp. 269-304.
B. G. Greenberg and A. E. Sarhan, "Matrix inversion, its interest
and application in analysis of data,"
JASA, Vol. 54 (1959),
pp.
£3_7
B. G, Greenberg and A. E. Sarhan, "Generalization of some results
for inversion of partitioned matrices ,."
in "Contributions to Probability and
To be published
Statistics~
Essays
in Honor of Harold Hotelling", Stanford University Press.
£4J
H. Hotelling, "Simplified calculations of principal components."
?sychometrika, Vol. 1 (1936) pp. 27-35.
£5]
S. N. Roy and A. E. Sarhan, "On inverting a class of patterned
matrices."
Biometrika, Vol. 43
(1956)~
pp 227-231.