UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill) N. C.
.,
A NOTE ON THE INEQUALITY OF THE CHARACTERISTIC ROOTS OF
T~
PRODUCT
AND THE SUM OF TWO SYMMETRIC MATRICES
by
C. G. Khatri
April 1964
Contract No. AF-AFOSR-84-63
This research was supported ~ the Mathematics Division of
the Air Force Office of Scientific Research.
Institute of Statistics
Mimeo Series No. 387
A NOTE ON THE INEQUALITY OF THE CHARACTERISTIC ROOTS OF THE PRODUCT
AND THE SUM OF TWO SYMMETRIC MATRICESl
by
C. G. Khatri
University of North Carolina and GUjarat University
= = = = = = ------- = = = = = = = = = = = = = = - - - - -
==
By asymmetric (sy.) matrix, we shall mean here a symmetric matrix
with real elements only.
vIe
establish the bounds on the i-th maximum
characteristic (ch.) root (denoted by
sy. matrix and B is any
rv
ch.)
of (i) AB where A is any
1
~
ruy
sy. positive semi-definite (p.s.d) matrix, and
(ii) of tv
A + B wbere A and B are both symmetric matrices.
rJ
I'V
Anderson
N
and Gupta [1], using somewhat different methods, established the bounds for
the case (i) when A was sy. p.s.d.
.-v
and B was
sy. p.d.
All the results
(V
established here are valid for hermitian matrices (idth possibly complex
elements) too.
'He require the follOidng three lemmas, which are given by Bellman [2].
I
t
I
Lemma 1:
Let A be any sy. matrix of order p
.eV
f
(j
!
Then
= 1,
2, .•• , i-l)
ch.A
11"
f
=
be column vectors with p
minimum
a.
-J
j
1, 2,
~.
maximum
Xl a .=0
=
... ,
-J
Xl
and let ! '
j
elements, real and finite.
~ ! )
Xl
(
E
X
i-l
Note that we replace the condition of
a~
a. = 1 (j = 1,2, ... , i-l)
- J-J
given by Bellman [2] for Courant-Fischer min-max. theorem by any a.
-.J
(j
=
1, 2, ... , i-l) having real and finite elements.
that by doing this, we are not changing the value of
It is easy to see
ch.A.
1",
~his research ivaS supported by the Mathematics Division of the
Air Force Office of Scientific Research.
2
Lemma 2:
=
Let ,-..,or
A
(a..
), i, j
~J
a selluenee of
sy. matrices.
Lemma 3:
~
nxp
Let
and
with n 2: p.
eh. .AB
~
NfV
~
= 1,
2, ... , r
= 1,
and r
2, .,., p be
Then
be two matrices of respective order pxn
and
Then
= eh.
~
for all positive eh. roots and
BA
,..,,..,
eh. AB = ch
. BA
~ "" '"
n-p+~
,..J
for all negative eh. roots.
tv
Let A be a sy. matrix of order p
Theorem 1:
and B, a sy. matrix of
tv
order p, be p.s.d. of rank
r
(~p).
Let
a
("
and b
denote respectively
the number of positive and the number of negative eh. roots of ,..J
AB.
......
for any
i,
j
= 1,2, ... , a
eh.B ch . jA < eh. AB < ehj
p+~- rJ
~
IN
(VN
and for
-
i = p - b + 1, P - b + 2, .•. , P
eh.B eh
J'"
Proof:
Since
~
,~
and
eh.
. 1 A
~-J+"'"
j
= 1,
roJ
B is sy. p.s.d., we ean find an orthogonal matrix
r"
-
(V
2, ... , r ,
. jA < eh. AB < eh.B ch . . ( l)A .
tV
~ ""'" J
r+J.-J- p- rV
r+~-
and WI > (J)2 > . .. > ill > 0
roots of B.
Then
Let
~
-
-
be a sub-matrix of order
are the nonzero eh.
r
r
6 sueh
I"J
of (6 A 6'), obtained
"" tV""
by deleting the last (p-r) rows and (p-r) columns of
Then by
lemma 3, it is easy to see that
= eh.SD
= eh.~ (nn-)
~"""'UJ
Sw ,.,S SIc)
(1)
eh.AB
(2)
ehp-r+~",
.AB = eh.~ (SD
)=eh.~ (nS n-)
rJrJlj)
oS!.:) ",,5Jo.>
J. "'IV
tv
where
Then by lemma 1,
for i
= 1, 2, ... , a and
for i=r-b+l, r-b+2, ... , r ,
3
=
ch.J. (SD
)
,..;,.Jt.;J
minimum.
9h
h
where!
and
9h
9h=0
= 1,2, ... ,i-l
are column vectors with
r
elements,
Now minimize over a sub-space given by
~i-j+h =
(0, ... ,0, ch,O, ... ,O),
ch
is nonzero finite and real at
h = 1,2, ... ,j-l, and ~ = (0, ... ,0, ~j,h" .. , ~,h)
the h-th place, for
~
!'
(h = 1,2, ... ,i-l)
real and finite.
for
maximum.
h = 1,2, ... ,i-j.
= 0.
= x 2 = ... = x j _l
°
for
Now let us write
r'
x' -J.-J+
a. . h =
Then
1
-
1
9..h = (w:J 2 e(.J, h""J w-"2(x
r r, h)
for
h
= 1,2,
h = 1,2, ... ,j-l
=
give
(JW.J x.,
... , J; x)
J
r r
... , i-j, ,El
be a sUb-matrix
of
S, obtained by deleting the first (j-l) rows and (j-l) columns of NS,
and
1?2
(W j '
= diag.
ill
(h = 1, 2, ... , i-j)
j +l ,···, wr ).
Note that the elements of
r
and
are finite and real, and the minimum over a sub-set
must always be greater than or equal to the minimum. over the larger set.
Hence (3) gives
(4)
<
ch.J. (SD)
fV('Ill
minimum
maximum.
6.
y'O =0
-n
- -h
h = 1,2, ... , i-j
-1
y'y
J --
Since
ill.
ch. (SD ) <
J.
,..J,.JW
or
minimum.
maximum
ih
r' 2.h=O
h = 1,2, ... , i-j
Using the lemma 2, and u. = ch. D , (5) gives
J
(6)
ch.J. (SD)
",,.J()J
ih
J""'G.)
< ( chJ.-J+
. . 1 S) (ch.J ,-Jill
D ) .
r-J
Sl)eJ.
( c h . . 1'"
J
J.-J+
J
•
4
Now consider
and
ch. SD > (Ch D ) (ch . .S)
j ""w
~ ~,.., w r+~-J"'"
Combining (6) and (7), and using lemma 2 for
(8)
for
(ch.B) (ch
J'"
= 1,
i, j
< ch. SD
. . A)
P+~-J
N
~
-
..... "'ill
S matrix, we get
tv
< (ch.B) (ch . . IA)
J ,...
-
~-J+'"
2, •.• , r
Now, the use of (1) and (2) in (8) proves the theorem 1.
Corollary
Let I"V
A be a sy. matrix,
l~
C be sy. p.d.
Then for any
tV
i, j
B be sy. p.s.d. of rank
rv
= 1,
2,
a (
"'J
r
and
= number of positive
ch. roots of AB) ,
~'"
(ch. BC)(ch
J "''''
and for
j
. . A C- l )
P+l-J
,-J
= 1, 2, ••• , r
l
< ch.J. AB < (ch.J BC)(ch.
. 1 AC- )
l-J+
oJ,.., -
..J
and
f'V""
NN
i = P - b + 1, P - b + 2, ... , p,
(b(= number of negative ch. roots of ~) ,
(ch. BC) (ch
J
Proof~
"V~
l
: ..AC- l ) < ch. AB < (Chj BC)(ch . . ( l)AC- )
J. ""Nr+J.-J- p-
r+l-J""""
rw-V
N-'-
Since £ is,sy. p.d., we have
C
= TT'
where T is a non-
Then by lemma 3,
ch.AB = ch.(AT,-l T' BTT- l ) = Ch.[(T-IA T,-l)(T'BT)]
singular matrix.
~.......,v
1",""-J
.--
"oJ_"""
1
r.J
"V,.y
""'"
and then by
.....,""0.1
theorem 1, we get the corollary 1.
Corollary 2:
i, j
= 1,
Let
r =p
in corollary 1.
Then, we get for any
< ch. AB <
l
(ch. BC) (ch . . 1 AC- ) .
2, ... , P ,
-1
. .AC )
P+l-J ........
(ch.BC) ( ch
J tv""
J.
rJ- -
J ""'...-
l-J+
"" .....
5
The above result was established by Anderson and Gupta [1] by using
somewhat different methods under the condition of r-J
A being
we have proved it for
p.s.d., but
A any sy. matrix.
(V
Corollary 3:
As a special case of corollary 2, we get for any
i = 1, 2, ... , p
max. [(ch BC)(ch.AC- l ), (ch.BC){
chp,....""
AC- l )] <
cb.1 (AB)
J,. ..... ,..
P"'N
rJ~
J,.",v
~ min [(Chl~S)(Chi~C:l), (Chi~q)(Chl~C~l)].
Theorem 2:
for any
Let
A and B be any two sy. matrices of order
= 1,2, .•. ,
i,j
J'"
Proof:
matrix ~
ill
p+l-J...... -
Since
~
> C)2->
J3 = 9 Bill
~
p, there exists an orthogonal
where,J<:J = diag. (WI' (;)2""
> ware
the ch. roots of B.
p
~
1 -
'V
= minimum
.
[(
maxJ.mum
x'u
-";;'h
=°
p)
,ill
and
Moreover, by lemma 1, we have
-
ch.1 (A+B)
,.J
.
.-.J
is a sy. matrix of order
such that
Then
P
. .A < ch. (A + B) < ch.B + ch. j lA
J...... "" JN
J.- +
ch.B + ch
p.
~'(A+B)~)
,
=
-xx-
h = 1,2, ... , i-I
Let us write
and
(h
= 1,
2,
... ,
i-I) .
Then
ch. (A+B) = minimum
J. '"
N
maximum
!h
~'J3h=O
h = 1,2, ... ,i-l
Now minimize over the sub-region given by
~'
tv
r
= 1,
.. =
h+J.-J
(0, •.• ,0, ch ' 0, ... ,0), ch
2, ... , j-l, and
Then -y' rvr-rl+J.~ . j
is nonzero finite and real, for
~h' = (0, ... ,0, ~. h' ... , ~ h) for h = 1,2, ... ,i-j.
'"
J,
r,
= ° (h = 1,2, ..• ,j-l)
give
Yl = Y2= ... = y.J- 1
= 0.
Now
6
let-us write Z
-
D~
N..;
= (y.,
... ,y),
J
r
_oh
=(
(3.
J,
~
h"'"
r,
h)(h
= 1,2, ... ,i-j),
= diag. (w.,
w.J+ l""'W)
and A be the sub-matrix of rJ
A, obtained by
J
p , . . . . ,l
deleting first (j-l) rows and (j-l)
columns of
.
(~'A~).
,.., rJ(V
With t his, it is
clear that
Z' A
ch.~ (A+B)
< minimum
rJ rJ
maximum
~h
~'~h=O
= 1,2,.,., i-j
h
< minimum
-
Z
"'1 Z'Z
L- -
+
(ch.~-J+
. 1 Al ) + w.
maximum
,..J
Z'5 =0
.§.h
J
- -h
= 1,2, ... , i-j
h
That is" using lemma 2,
< ch.B + ch. . lA
ch.(A+B)
~
"" ""
J.v
For the other part, ch.B
IN
But
ch. "l(-A)
J -b"
-v
=-
ch
~-J+
= ch.[(A+B)
+ (-A)] < ch.(A+B) + ch . . l(-A) .
J'V'~
,."
~
J-~+.v
<'V
. . A.
""
P+~-J
N
~
-
'"'
Hence we get
ch. (A+B) > ch.B + ch
~
r.J
J rJ
. .A
P+~-J
,...
Thus, the theorem 2 is established.
Corollary 4:
As a special case of theorem 2, we have for any
i
= 1,2" ... ,p
max. [ch.B
+ chpt.J
A , chpry
B + ch.A'
< ch.(A+B)
< min. [chlB+ch.A"Ch.B+ChlA].
~'"
~~ ~ ro.---- ~,..,
1.""
""
The author thanks Professor S. N. Roy for his kind help.
REFERENCES
(1] Anderson, T. W. and S. Das Gupta (1963), Some inequalities on characteristic roots of matrices, Biometrika, Vol. 50, PI'. 522-524.
[2]
Bellman,
(1960), Introduction to Matrix Analysis, McGraw Hill Book
Co., New York.