Nonnegative definite hermitian matrices with increasing principal minors

0.07 MB·English

by  Shmuel Friedland

Nonnegative definite hermitian matrices
with increasing principal minors
Shmuel Friedland∗
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1 January 20, 2013
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Abstract
a
J
Anonnegativedefinitehermitianm×mmatrixA6=0hasincreasingprinci-

0 pal minors if detA[I]6det[J] for I ⊂J, where detA[I] is the principal minor
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of Abasedonrowsandcolumns inthe setI ⊆{1,...,m}. For m>1 we show

] A has increasing principal minors if and only if A−1 exists and its diagonal
A entries are less or equal to 1.
C
2010 Mathematics Subject Classification. 15B57, 90C10
.
h
Key words. Submodular functions, Hadamard-Fischer inequality, CUR approxi-
t
a
mations.
m
[
1 Introduction and statement of the main result

1
v
5 Let m be a positive integer and denote [m] := {1,...,m}. A real valued function
6 w : 2[m] → R defined on all subsets of [m] is called nondecreasing if w(I) 6 w(J)
6
4 when I ⊂ J ⊂ [m]. It is submodular if
.
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0 w(I)+w(J) >w(I ∪J)+w(I ∩J)
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1 for any two subsets I,J of [m]. The importance of submodular functions in com-
:
v binatorial optimization is well known. Several polynomial time algorithms to min-
i
X imize a submodular function under a matroid constraint are known, we refer the
r reader to the survey [Iwa08] for more information. The maximization of a sub-
a
modular function under a matroid constraint, and specially, under a cardinality
constraint, ν (w) := max w(I), is also of great interest. For some submod-
k I⊂[m],|I|6k
ularfunctionsw thelatter problemisNP-hard. However, aclassical result[NWF78]
shows that when w is nondecreasing and submodular, the greedy algorithm allows
one to compute an approximation νG(w) of ν (w) which is such that νG(w) >
k k k
(1−e−1)ν (w).
k
Denote by H ⊃ H the cone of m × m nonnegative definite hermitian
m,+ m,++
matrices and its interior consisting of positive definite hermitian matrices respec-
tively. For I ⊆ [m] denote by A[I] the principal submatrix of A, obtained from A
∗DepartmentofMathematics,StatisticsandComputerScience,UniversityofIllinoisatChicago,
Chicago, Illinois 60607-7045, USA, [email protected]. This work was supported by NSF grant
DMS-1216393.
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by deleting the rows and columns in the set [m]\I. Recall that the principal minors
of a nonnegative definite matrix satisfy the multiplicative submodularity property:
detA[I ∪J]detA[I ∩J]6 detA[I]detA[J], where I,J ⊆ [m], A ∈ H . (1.1)
m,+
We assume here that detA[∅] = 1. In other words, the function log(·,A) : 2[m] → R
given by
log(I,A) := logdetA[I], I ⊆ [m], A ∈H (1.2)
m,+
issubmodular. Thisinequalityhasarisenintheworkofseveralauthors. Itgoesback
to Gantmacher and Kre˘ın [GK60] and Kotelyanski˘ı [Kot50], see the discussion by
Ky Fan [Fan67, Fan68]. The classical Hadamard-Fischer inequality for the principal
minors of nonnegative definite matrices is obtained when I∩J = ∅. It is well known
that the inequality (1.1) hold also for M-matrices, e.g. [Car67].
[FG12, §5] discusses the CUR approximation [GTZ97] of nonnegative definite
hermitian matrix. The main problem there is to find a good approximation to the
maximum of detA[I] on all subsets I of [m] of cardinality k. Assuming that A
has increasing principal minors the greedy algorithm is applied to give an estimate
for the CUR approximation. It is shown in [FG12] that if all eigenvalues of A are
greater or equal 1 then A has increasing principal minors. The purpose of this note
is the following theorem.
Theorem 1 Let A ∈ H \{0}. Assume that m > 1. Then A has increasing
m,+
principal minors if and only if A is positive definite and all diagonal entries of A−1
are less or equal to 1.
2 Proof of Theorem 1
Assume first that A has increasing principal minors. Suppose that detA = 0.
Since A is nonnegative definite we have that 0 6 detA[I] 6 detA = 0 for any
nontrivial subset I of [m]. Hence A = 0 contrary to our assumption. Therefore
A is positive definite. Let B = [b ] := A−1. Clearly B is positive definite. As
ij
A has increasing principal minors we deduce that detA[[m]\{i}] 6 detA. Hence
b = detA[[m]\{i}] 6 1 for each i ∈ [m].
ii detA
It is left to show that if A is positive definite, B = [b ] := A−1 and b 6 1
ij ii
for i ∈ [m] then A has increasing principal minors. We first observe that B has
decreasing principal minors, i.e detB[I]> detB[J] if I ⊂ J. Indeed, it is enough to
consider the case where J = I ∪{j}, where j ∈/ I. Then the Hadamard-Fischer in-
equality yields detB[J] 6 b detB[I] 6 detB[I]. Recall the Sylvester determinant
jj
detA[I]
identity: detB[[m]\I] = . Since B has decreasing principal minors it follows
detA
that A has increasing principal minors. 2
Corollary 2 Let A ∈ H and m > 1. Denote B = [b ] := A−1. Then
m,++ ij
tA,t > 0 has increasing principal minors if and only if t > max b .
i∈[m] ii
2
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