•
A USEFUL THEORJ.'1'I IN HATRIX ALBEGRA
by
S. N., Roy
Institute of Statistics
University of North Carolina
Institute of
Stati~tics
lVIimeopraph Series T'o. h7
December, 1953
A USEFUL THEOREM IN i"JIATRIX
,~LGEBRA
by
S. N. Roy, Institute of Statistics, University of North Carolina
1.
Introduction and notation.
In this paper a theorem on the
characteristic roots of (square) matrices, believed to be new,
tained, which has
is ob-
proved extremely useful in a sector of mathematical
statisticJ and Which, the author hopes, might be useful in other sectors
of applied mathematics.
It will be assumed here that the elements of the
matrices considered are real or complex numbers.
Any capital letter, say,
Mwill stand for a matrix, M for its transpose, ~r~ for its conjugate
f
transpose, m.. for its (ij)-th element,
~
m~
..
for the conjugate of m.
0' and
~
M(p x q) will denote that the matrix consists of p rows and q columns.
In this paper, so far as the new results are concerned, only square matrices
will be discussed for which we have p
= q~ 1. For any matrix M, the rank
will be denoted by r(M), any characteristic root by c(M).
If M is p x p,
there will be, of course, p such roots, say, cl ' c 2 , ••• , cpo I(p) will
stand for a p x p unit or identity matrix and D (p) will stand f~r a
A
p x p diagonal matrix whceediagonal elements are, say, AI' A2 , ••• , Ap '
In course of this paper, some well known results in matrix algebra will
- -7,
be used /-1, 2
(1.1)
including the following:
The set ci(AB)(i=l, ••• ,p)
matrices.
= Ci(BA), where A and B are two p x p
It may be noted that this result can be easily generalized to
2
meaning that any non-zero root of AB is a non-zero of BA and vice versa.
Notice that, out of the q c(BA)IS, q-p must be necessarily zero o
(1.3)
If M is a p x p hermitian and positive-definite or positive-semi-
definite matrix (to be called respectively pod. or p.s.d.), i.e., if all
c(M)Js are positive or non-negative, then there is a unitary matrix P such
_
(1.4)
~f-
-7 = r/- A (q
r;-A(p x q)
-
-7 = r(AA
x p)
'3{-
)
c
~<
(1.5)
A(p x q)A"(q x p) is herm:i.tian and at least pos.d., so that all
C(AA~~) will be at least non-negative.
will be actually used in this paper
0
The case of q
= p is the one that
2. 1.
~
bound theorem
~
characteristic roots _ If A and B are two
p x p matrices of "Thich at least one is non-singular, than for all the
characteristic roots c(AB) we have
,
(2.1. )
where c. and C
stand respectively for the smallest and the largest
mm
max
characteristic root (each, of course, non-negative).
Proof.
By (1.3) there are unitary matrices, say FA and P such that
B
Not:tce that all C(AA~!-) and
c(BB-l!-) are real and non-negative and, if we
further assume that A is non-singular, then all c(AA-l!-) are positive.
From (2:1.2) it follows that it is possible to find unitary
matrices, say QA and QB,such that
We have now, for any characteristic
r~0t
of AB,
4
(?1.4)
Notice that, since, PA' PB, QA' Q are all unitary, therefore, QAPB and
B
~PA
are also each unitary.
Now, i f c is to be a characterist.ic root of AB, there exists set
of (complex) numbers zl' ••
0
'
0
zp' not all of lolhich are zero, such that
the following set of equations are satisfied.
(2.1.5)
P
2:
A.r ..fl..s'kzk = cZ i (i...1, 2,
j,k=l 1 1J J J
.0"
p).
Remembering that A:1S and fl..IS are real, and (here) fl..IS are non-nega1
J
J
tive and Ai Is positive,iand dividing by Ai) and taking the conjugate of
(2.~.5),
multiplying the two and then summing over i~1,2, ••• , p, we
have
(2.1.6)
i
:;; cc-
~
i
Z.z.
I A.2
1 1/ 1
eo
Now, since R is unitary, we have
Zr i
i
.r..
J lJ
I
=
OJ J' t (where
° is the Kronecker symbol) •
Thus (2.1.6) reduces to
- / A.2 = i:
2
- I
cc.tz.z.
lJ..s··ks·kIZkzl
i l l/
l
j,k~kl J J J
K
(2.1.8)
•
It is easy to see that the coefficients of l/A~ls on the left-hand
l
2
side and those of 1J..ls on the right hand side are each at least nonJ
ne ga tive • Hence, if we replace all IJ.. IS by IJ.
and a 11 A. I S by Amax '
max
1
the right hand side is increased (or at least not dlininished) and the
J
left hand side is d:iminished (or at least not increased). lIe have thus
~
2
i.e. , -< IJ.max
Z °kklZkZk 1
j
i.e.,
Since
<
-
2
f..l.max
-
Z z.z.
i
1
l
•
ZZ'Zi is positive, it follows that
.
l
l
(since S is unitary)
,
6
cc <
)..2
2
max f..l.max
-
(2.1.10 )
'
~r
i~
c(AB)c(AB) < c
(AA)c
(BB)
- max
max
•
Likewise, in (2.1.8), replacing all A,'s by ~,
ilun
~
and all
fJ...'s
by
fJ..' and arguing in a similar manner, we have
J
. m~n
(2.1.11)
•
Combining (?l.lO) and (2.1.11), we have the theorem
(2.1.1).
It is
easy to see that (2.1.1) can be generalized to the case of the product
of any finite number of matrices AI' A , ••• , An' provided that not
2
more than one of them is possibly singular, the rest being all nonsingular.
Thus we have the theorem.
n
n
n
n
T{'
, (Ai'A~~) ~ cCIT
A,) <)t c ax(A,A~:-).
, e..u.u
m.,.......
, 1Ai)cCR
' 1 ~ - ~=
. 1 m
:1: ...
.,
~...
~=
~=
1
2.2.
~
useful special cases of (2.1.1).
Putting A ~ I(p) in (2.1.1) we have
(??.l)
- < c (AA)
*
cmln
. (AA ~~ ) -< c(A)c(A)
- max
- -7
a result due to Professor E. T. Browne /-1
•
,
7
If A and B are both hermitian, then it is well known that all
c(A)ts and C(B)fS are real and also that
Hence, in this case, (2.1.1) will reduce to
•
Suppose that A and B are both hermitian and one of them, say A,
is ped. and the other, i.e., B, at least p.s.d.
Then all c(A)t s are
real anct positive, all C(B)1S real and at least non-negative.
Also
AB is at least p.s.d., so that all c(AB) are real aud at least nonnegative.
In this case, (2.1.1) reduces to
(A)c
(n)
cmin(A)c . (B) < c(AB) < c
ffiln
- max
max
•
REFERENCES
1. E. T. Browne, Bull Amer. Math. Sec. Vol. 34(1928), pp. 363-368.
....-.-
2.
.
C. C. Macduffee, Thg
New York, 1946,
-.c-.
-"~
~heory
W.
of-Matrices, Chelsea Publishing Co.,
17-~.