CommuniCllt;ons of the Moscow MathemtltlCllI Society
225
A multiplicative analogue of Bergstrom's inequal ity for
the Hadamard matrix product
A.D. Berenshtein and A.D. Vainshtein
1. Let A and B be symmetric positive-definite matrices, and Ai and Bi the submatrices obtained by
deleting the ith row and ith column. The following is the well-known Bergstrom inequality (see
[1) -[3]):
(t)
This result can be reformulated as follows.
Theorem 1. Let A and B be symmetric positive-definite matrices; then for any x and y
(2)
(A-Ix, x)
Proof It is easy to verify that
y) ~
( M-I x x)
(3)
where
+ (B-I y,
,
/.I.
«A + B)-I (x+y),
(x+y».
=11_1 ~IMI'
~I
r
is an arbitrary number. Writing the three scalar products in (2) in the form (3), we obtain an
inequality similar to (1) (for any fixed x the block matrix
(~ ~) can be made positive-definite by
choosing a sufficiently large /.I.). Thus (2) follows from (1).
We now prove the converse. The relation (3) for a symmetric positive-definite n x n matrix T can
be rewritten in the form IT 1/1 Tn 1= tnn - (Tnl~,tn},where ~ = (tIn' t 2n , .•. , tn-l,n)'
Replacing the three terms in (1) by such expressions we obtain an inequality similar to (2). Thus (2)
is a reformulation of (1).
Remark. Another proof of (2) using quasi-linearization was given in [4].
2. Let P and Q be arbitrary matrices of the same size. The Hadamard product of P and Q is the
matrix R = P*Q, where rij = PilIi} (see [2]). As is evident from Theorem 1, the following theorem
can be regarded as a multiplicative analogue of Bergstrom's inequality.
Theorem 2. Let A and B be symmetric positive-definite matrices; then for any x and y
(4)
Proof We introduce the square matrix Y = (Yij), where Yij
=
YiYj- Then (4) can be rewritten as
(A-IX, x) • (B-1y, y) ~ «A.B)-I.Y x, x).
We shall write M
written in the form
~ ]V
if the matrix M - N is non-negative definite. Then the last inequality can be
(5)
By well-known results on the mono tonicity of matrix functions ([ 3], 464), M ;;;, ]V > 0 is
equivalent to N- I ;;;, M-I > O. Since A and Bare positive-definite, by Schur's theorem ([3], 258)
A*B is also positive-definite. Hence it is easy to see that if all the Yi are non-zero, then both sides of
(5) are positive-definite. Consequently (5) is equivalent to
(6)
Y,
It is not difficult to verify the relation (M-l • Y)-l = 1If.
where Y is the inverse of Y with
«B-ll/, I/)B - Y)~ O.
respect to Hadamard multiplication. Substituting this in (6) we get A.
By Schur's theorem already mentioned it is now sufficient to show that (B-Iy, y)B - Y ~ 0, or
Y ..
(JJ-1Yt y) • (Bx, x) ~ (Y x, x) = (Yt Z)2 •
226
Communications of the Moscow Mathematical Society
The last inequality is called the generalized Bergstrom inequality and is proved in [1], 69. Hence
if all the Yi are non-zero, (4) is true. In the case when some components of y vanish it is sufficient to
use a passage to the limit.
3. Inequality (4) enables us to obtain easily the majority of the known inequalities concerning the
.determinants of Hadamard products. In the following A and B are always positive-definite symmetric
matrices.
Corollary 1 (Oppenheim, [5J, 421).
IA
(7)
*B I ~
~ A
I
bJl X : .• X b"JI'
Proof Using (3), it easily follows from (4) that
IAI ) (
( ann -lA,;'I
bnn -
IBI )
IBnl ~ annbn n -
IA*BI
IAn*Bnl •
Hence, as B is positive-definite, it follows that
IAI )
( ann- IAnl
bnn ~annbnn or, equivalently,
by induction.
IA • B I/ IA I
~ bnn
* Bn I / I All
I An
IA • B I
Corollllry 2 (Oppenheim, [3J, 258).
IA.BI
IAn.Bnl
~
,. (7) now easily follows from this
lAB ,.
The proof follows from (7) and the well-known inequality
(see [IJ, 63).
I B I <=
bl l X . .• X bn n
Corollary 3 (Schur, [2J, 121).
(8)
I A • B I + I AB I ~ I A I bu
X ••• X bnn
+ I B I au X
••• X
ann'
Proof Without loss of generality we can assume that all the diagonal elements of A and B are equal
to 1. Then (8) is equivalent to
(9)
IA
* B I + I AB I ~ I A I + I B
I,
and (4), by using (3), is equivalent to
(10)
Now (9) is easily obtained from (10) by induction by using the inequalities
I B I < I Bn I < 1, I A *B I <= I An .Bn 1<=1.
I A I < I An I <=
1,
References
[1] E. Beckenbach and R. Bellman, Inequalities, Ergeb. Math. Grenzgeb. 30, Springer-Verlag, BerlinNew York 1961. MR 28 # 1266.
Translation: Neravenstva, Mir, Moscow 1965. MR 31 # 5937.
[2] M. Marcus and H. Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon,
Boston, MA, 1964. MR 29 # 112.
Translation: Obzor po teorii matrits', matrichnykh neravemtv, Mir, Moscow 1983.
[3 J A. W. Marshall and I. OIkin, Inequalities: theory of majorization and its applications, Math. in
Science and Engineering 143, Academic Press, New York-London 1979. MR 81b:00002.
Translation: Neravenltva: teoriya mazhorizatlii i ee prilozheniya, Mir, Moscow 1983.
MR 85a:00004.
[4] A.D. Vainshtein, Optimal extrapolation of a random flow in a tree-like network, in:
Ralpredelennye systemy peredachi i obrabotki informatsii (Distributed transmission systems and
information processing), Nauka, Moscow 1985, 120-128.
[5J L. Mirsky, An introduction to linear algebra, Clarendon Press, Oxford 1955. MR 17-573.
Moscow Institute of Railway Transport
Engineers
Received by the Board of Governors 1 April 1986