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ARS MATHEMATICA CONTEMPORANEA
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 9 (2015) 261-266
On mixed discriminants of positively definite matrix*
Chang-Jian Zhao f
Department of Mathematics, China Jiliang University, Hangzhou 310018, P. R. China
Xiao-Yan Li
Department ofMathematics, Hunan Normal University, Changsha 410000, P. R. China
Received 2 November 2012, accepted 21 February 2014, published online 11 January 2015 Abstract
In the paper, some new inequalities for the mixed discriminants of positively definite matrix are established, which are the matrix analogues of inequalities of the well-known mixed volumes function.
Keywords: Mixed discriminants, symmetric matrix, mixed volume, Aleksandrov's inequality. Math. Subj. Class.: 15A09, 52A40
1 Introduction
Let xi,... ,xn be a set of nonnegative quantities and let Ei(x) be the ¿-th elementary symmetric function of an n-tuple x = x(xi,..., xn) of non-negative reals is defined by
E0(x) = 1 and
Eyj ('x) — ^^^ 'x j i 'x • • • 'x j ■ , 1 ^^ i ^^
i<jl<-<ji<n
An interesting inequality for the symmetric function was established ([1], also see [2], p. 33) as follows.
Ei(x + y) Ei(x) Ei (y)	(11)
Ei-i(x + y) - Ei_i(x) + Ei-i(y).	(.)
* Research is supported by National Natural Science Foundation of China (11371334). t Author to whom correspondence should be addressed.
E-mail addresses: chjzhao@163.com chjzhao@aliyun.com chjzhao315@sohu.com (Chang-Jian Zhao), lixy-77@163.com (Xiao-Yan Li)
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
262	Ars Math. Contemp. 9 (2015) 165-186
A matrix analogue of (1.1) is the following result of Bergstrom [3].
Let K and L be positive definite matrix, and let K and Lj denote the sub-matrices obtained by deleting the i-th row and column. Then
det(K + L) > det(K) + det(L)
det(Kj + Lj^ _ det(Kj) det(L,)'
An interesting proof is due to Bellman [4] (also see [2], p. 67). A generalization of (1.2) was established by Ky Fan [5] (also see [6-7]). Moreover, we assume all positive definite matrix are supposed to be symmetric in the article.
There is a remarkable similarity between inequalities for symmetric functions (or determinants of symmetric matrices) and inequalities for the mixed volumes of convex bodies. In 1991, V. Milman asked if there is version of (1.1) or (1.2) in the theory of mixed volumes and it was stated as the following open question (see [8]):
Question 1.1. For which values of i is it true that for very pair of convex bodies K and L
in Rn,
Wj(K + L) Wj(K) + Wj(L) ? Wj+i(K + L) > Wj+i(K) + Wj+i(L) "	( ' )
The convex body is the compact and convex subsets with non-empty interiors in Rn. Wj(K) denotes the quermassintegral of convex body K and Wj+i (K) denotes the mixed volumes V(K,..., K, B,..., B). The sum + is the usual Minkowski vector sum and B
n — i—1	i+1
denotes the unit ball.
A theorem by Minkowski provides a fundamental relation between volume and operations of addition and multiplication of convex bodies by nonnegative reals: If K1,..., Km are convex bodies, m G N, then the volume of t1K1 + • • • + tmKm is a homogeneous polynomial of degree n in tj > 0 (see [14]). That is
V(t1K1 + ••• + tmKm) = £ V(Ki! ,...,Kin )t ji ••• tin ,
1<ii !...Jjn<m
where the coefficients V(Kii,..., Kin) are chosen to be invariant under permutations of their arguments. The coefficient V(Kii,..., Kin) is called the mixed volume of the n-tupe (Kii,..., Kin). Steiner's formula is a special case of Minkowski's theorem; the volume of K + tB, t > 0, can be expanded as a polynomial in t:
n
n
V(K + tB) = Y (. JWj(K)tj
i=0
where Wj(K) := V (K,..., if, B,..., B) is the quermassintegral of convex body K.
n— j	j
A partial answer (L must be a ball) of (1.3) was established by Gianopoulos, Hart-zoulaki and Paouris [9]).
If K is a convex body and D is a ball in Rn, then for i = 0,..., n — 1
Wj(K + D) Wj(K) + Wj (D) Wj+1(K + D) > Wj+1(K) + Wj+1 (D)
(1.4)
C.-J. Zhao andX.-Y. Li: On mixed discriminants of positively definite matrix
263
The answer to the above question is negative; it can be proved that (1.3) is true in full generality only when i = n — 1 or i = n — 2 (the details see [10]). Moreover, a dual inequality of (1.4) for the dual quermassintegral of star bodies was proved by Li and Leng [11].
In the paper, we establish some inequalities for mixed discriminants of positively definite matrix which are matrix analogues of some mixed volumes inequalities.
2 Mixed discriminants and Aleksandrov's inequality
Recall that for positive definite n x n matrices K1,..., KN and X1,... ,XN > 0, the
determinant of the linear combination AiKi +-----+ XNKN is a homogeneous polynomial
of degree n in the Ai (see e.g. [12]),
det(AiKi + ••• + An Kn )= ^ D(Kit ,...,Kin )Ail ••• Ain, (2.1)
1<ii,......,in<N
where the coefficient D(Kil,..., Kin) are chosen to be invariant under permutations of their arguments. The coefficient D(Kil,... ,Ki ) is called the mixed discriminant of Kil ,...,Kin .
The mixed discriminant D(K,..., K, I,... ,I), with n — k copies of K and k copies of the identity matrix, I, will be abbreviated by Dk (K). From (2.1), we have
n-i _ -
Dj(K + AI) = ^ AjDi+j(K).	(2.2)
j=0 j
Note that the elementary mixed discriminants D0(K),..., Dn(K) are thus defined as the coefficients of the polynomial
n
det(K + AI) = ^ Q AiDi(K).	(2.3)
i=0 i
Obviously, Do(K) = det(K) while nDn-i(K) is the trace of K.
The well-known Aleksandrov's inequality for mixed discriminants can state as follows (see [13], also see [14], p.383 or [15], p.35):
Lemma 2.1. If Ki,K2,..., Kn are real symmetric positively definite n x n matrices, then D(Ki, K2,K3, ...,Kn)2 > D(Ki,Ki, K3,..., Kn)D(K2, K2, K3,..., Kn), (2.4) with equality if and only if Ki=AK2 with positive number A.
3 Inequalities for mixed discriminants of positively definite matrix
Theorem 3.1. Let K be symmetric positively definite matrix and I stands for the identity matrix and t > 0. If 0 < i < n — 1 and i G N, then the function
Di(K + tI)
9(t) = D+ i(K + tI)	(3J)
is an increasing and concave function on [0, +to).
264
Ars Math. Contemp. 9 (2015) 165-186
Proof. If /¿(s) = A(K + si), then by the linearity of the mixed discriminant we see that
+ £) = E
j=0
£j Di+j (K + si)
Hence
Similarly, we obtain
= /i(s) + e(n - i)/i+i(s) + o(e2).
d/i(s) = lim /(s + £ - /(s) ds	£
= (n - i)/i+1(s).
d/i+i(s) ds
= (n - i - l)/i+2(s).
From (2.4), we obtain for all 0 < i < n
/i+i(s) - /i(s)/i+2(s) > 0,
with equality if and only if K =
From (3.2), (3.3) and (3.4), we have
d/i(s) ; ^ A d/i+1(s)
—;—/i+i(s) - /(s)-1-
ds	ds
(3.2)
(3.3)
(3.4)
Therefore
= /2+1 (s) + (n - i - 1)(/i+i(s) - /i(s)/i+2(s))
> /i+i(s). dg(s) = / /i(s) V
ds V /i+i(s)y
//(s)/i+i (s) - / (s)/i+i(s)
/i+i(s)
= (n - i) - (n - i - 1)
j"»(s)/»+2(s) /2+i(s) .
(3.5)
Hence
/ (t) =
Di(K + ti)
Di+i(K + ti) is an increasing and concave function on [0, +to).
□
Theorem 3.2. Let K be symmetric positively definite matrix and I stands for the identity matrix. If 0 < i < n, then
(n - i)Di+2(K)(Di+i(K)2 - Di(K)A+2(K))
> (n - i - 2)Di(K)(Di+2(K)2 - A+i(K)A+s(K)). (3.6)
n—i
j
C.-J. Zhao andX.-Y. Li: On mixed discriminants of positively definite matrix
265
Proof. Let / (t) = + tI) for t > 0 and g(t) = /i(t/L, then
fi+i(t)
i+i
dff(*)_, , . 1/i(i/+2(t)
f 2 'i+1
_ = (n - i) - (n -. - 1) /+ .	(3.7)
By differentiating the both sides of (3.6) again, we have
^ = -<» - • -1)
(n - i/i+2(t)/i+i(t) + (n - i - 2)/i(t)/i+i(t)/i+3(t) - 2(n - i - 1)/i(t)/+2(t)
x	/3+i (t)	.
(3.8)
From (3.8) and in view of g(s) being a concave function, we obtain
(n - i/i+2(t)/+i(t) + (n - i - 2)/i(t)/i+i(t)/i+3(t) - 2(n - i - 1)fi(t)/+2(t) > 0,
for t € (0, +to).
This can be equivalently written in the form
(n - i/i+2(t)(/+i(t) - /i(t)/i+2(t)) > (n - i - 2)/i(t)(/i+2(t) - /i+i(t)/i+3(t)).
(3.9)
Hence
(n - i)Di+2(K + tI) (Di+i(K + tI)2 - Di(K + tI)A+2(K + tI))
> (n - i - 2) A(K + tI) (Di+2(K)2 - A+i(K + tI)A+3(K + tI)). (3.10)
Notice that /i(t) is continuous function, letting t ^ 0+ in (3.10), (3.10) reduces to the
inequality in Theorem 3.2.	□
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