Cone-theoretic generalization of total positivity

0.29 MB·English

by  O. Y. Kushel

Cone-theoretic generalization of total positivity
O.Y. Kushel
Institut fu¨r Mathematik, MA 4-5, Technische Universita¨t Berlin, D-10623 Berlin,
3 Germany
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Abstract

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1 This paper is devoted to the generalization of the theory of total positivity.

We say that a linear operator A : Rn → Rn is generalized totally positive
]
P (GTP), if its jth exterior power ∧jA preserves a proper cone K ⊂ ∧jRn
S j
for every j = 1, ..., n. We also define generalized strictly totally positive
.
h
(GSTP) operators. We prove that the spectrum of a GSTP operator is pos-
t
a
itive and simple, moreover, its eigenvectors are localized in special sets. The
m
existence of invariant cones of finite ranks is shown under some additional
[
conditions. Some new insights and alternative proofs of the well-known re-
1
sults of Gantmacher and Krein describing the properties of TP and STP
v
1 matrices are presented.
3
7 Keywords: Cones of rank k, Cone-preserving maps, Gantmacher–Krein
3
theorem, Total positivity, Compound matrices, Exterior products.
.
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2000 MSC: Primary 15A48, Secondary 15A18, 15A75
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:
v 1. Introduction
i
X
The theory of totally positive matrices and kernels started with Kellog
r
a [9, 10] and mainly developed in monographs [5] by Gantmacher and Krein
and [8] by Karlin, nowadays becomes an interesting and important part of
the modern analysis. A matrix A is called positive if all its elements a
ij
are positive. A n × n matrix A is called strictly totally positive (STP) if
its jth compound matrix A(j) is positive for every j = 1, ..., n. (Recall
i ... i
that A(j) is the matrix that consists of all the minors A 1 j , where
k ... k
1 j
(cid:18) (cid:19)
Email address: [email protected](O.Y. Kushel)
Preprint submitted to Elsevier January 17, 2013
1 ≤ i < ... < i ≤ n, 1 ≤ k < ... < k ≤ n, of the initial matrix A. The
1 j 1 j
minors are listed in the lexicographic order. The matrix A(j) is n × n
j j
n!
dimensional, where n = . The first compound matri(cid:0)x(cid:1)A(1)(cid:0)i(cid:1)s
j j!(n−j)!
equal to A). (cid:0) (cid:1)
We introduce the following definition which gives a natural generalization
of the class of STP matrices. Given a family of proper cones {K , ..., K },
1 n
Kj ⊂ R(nj), we call a n × n matrix A generalized strictly totally positive
(GSTP) with respect to {K , ..., K } if its jth compound matrix A(j)
1 n
maps K \{0} into int(K ) for every j = 1, ..., n.
j j
The following result of Schoenberg is known for STP matrices (see, for
example, [15]). Let us recall the following two ways of counting for the
numberofsignchangesofavectorx = (x1, ..., xn) ∈ Rn. S−(x)denotesthe
number of sign changes in the sequence (x1, ..., xn) of the coordinates with
zero terms discarded. S+(x) denotes the maximum number of sign changes
in the sequence (x1, ..., xn) where zero terms are arbitrarily assigned values
±1 (see, for example, [15], p. 76).
Theorem 1 (Schoenberg). Let a n × n matrix A be STP. Then the fol-
lowing inequality holds for each nonzero vector x ∈ Rn:
S+(Ax) ≤ S−(x).
We construct special sets T(K ) ⊆ Rn with respect to the cones K ⊂
j j
R(n)
j . ThusweobtainthefollowinggeneralizationoftheSchoenberg theorem.
Theorem 25. Let a n×n matrix A be GSTP with respect to {K , ..., K }.
1 n
Then the interior of the set T(K ) is nonempty and the inclusion x ∈ T(K )\
j j
{0} implies the inclusion Ax ∈ int(T(K )) for every j = 1,..., n.
j
We also generalize the classical Gantmacher-Krein theorem (see, for ex-
ample, [1, 15, 16]) to the case of GSTP matrices.
Theorem 2 (Gantmacher, Krein). Let the matrix A of a linear operator
A : Rn → Rn be STP. Then all the eigenvalues of the operator A are positive
and simple:
ρ(A) = λ > λ > ... > λ > 0.
1 2 n
2
The first eigenvector corresponding to the maximal eigenvalue λ is strictly
1
positive and the jth eigenvector x corresponding to the jth in absolute value
j
eigenvalue λ has exactly j − 1 changes of sign. Moreover, the following
j
inequalities hold:
p p
q −1 ≤ S−( c x ) ≤ S+( c x ) ≤ p−1
i i i i
i=q i=q
X X
p
for each 1 ≤ q ≤ p ≤ n and c2 6= 0.
i
i=q
P
We construct special sets T(K ,...,K ) ⊆ Rn, j = 1, ..., n with respect
1 j
to the given family of cones {K1, ..., Kn}, Kj ⊂ R(nj).
Theorem 22. Let a linear operator A : Rn → Rn be GSTP with respect to
a totally positive structure {K , ..., K }. Then all the eigenvalues of the
1 n
operator A are positive and simple:
ρ(A) = λ > λ > ... > λ > 0.
1 2 n
Thefirst eigenvectorx correspondingto the maximaleigenvalueλ belongsto
1 1
int(K ) and the jth eigenvector x corresponding to the jth in absolute value
1 j
eigenvalue λ belongs to int(T(K ,...,K ))\T(K ,...,K ). Moreover, the
j 1 j 1 j−1
following inclusions hold:
p
c x ∈ int(T(K , ..., K ))\T(K , ..., K )
i i 1 p 1 q−1
i=q
X
for each 1 ≤ q ≤ p ≤ n and c 6= 0;
p
p
c x ∈ T(K , ..., K )\T(K , ..., K )
i i 1 p 1 q−1
i=q
X
for each 1 ≤ q ≤ p ≤ n.
The organization of this paper is as follows. In Section 2, we introduce
basic definitions concerning exterior powers of finite-dimensional spaces. In
Section 3, we recall basic definitions of the theory of cones and provide some
examples. Here we also give the definition of a cone of a finite rank. Section
3
4 deals with a certain duality between cones of rank j in Rn and proper
cones in its jth exterior power ∧jRn. In particular, we construct a special
set T(K ) ⊂ Rn for a given proper cone K ⊂ ∧jRn and study its topological
j j
properties. We conclude that under certain additional assumptions the set
T(K )isaconeofrankj. InSection5, weconstruct aset T(K ,...,K )with
j 1 j
respect to a family of proper cones K ⊂ Rn, K ⊂ ∧2Rn, ..., K ⊂ ∧jRn.
1 2 j
We give examples of such sets and study their topological properties. In
Section 6, we state the main results of the theory of cone-preserving maps. In
Section 7, we recall basic facts concerning exterior powers of linear operators
in Rn. In Section 8, we introduce the concepts of generalized total positivity
(with respect to a given family of proper cones), generalized strict total
positivity and generalized sign-regularity. Such definitions provides natural
generalizations of the classes of totally positive, strictly totally positive and
sign-regular matrices, respectively. Basic properties of GTP, GSTP andGSR
operators are listed in Section 9. The results of this section shows that
the class of GSR operators covers the entire class of operators with real
spectrum. InSection10, westateandprovethegeneralizationoftheresult of
Schoenberg concerning variation-diminishing properties of SR matrices. The
results of this section shows that a GSR (with respect to a family of proper
cones) operator preserves conic sets constructed as it was shown in Sections
4-5. Our main result concerning spectral properties of GSTP operators is
proved in Section 11. In this section we also provide some conditions for a
family of cones which are necessary for the existence of at least one GSTP
operator. Then we state and prove a stronger statement describing invariant
sets of a GSSR operator. In Section 12, we deduce the classical results on TP
and STP operators (which are special cases of GTP and GSTP operators)
from the preceding reasoning. In Section 13, we study one more special
case of GTP matrices, in particular, matrices every compound of which is
diagonally similar to a positive matrix. We list some special properties of
such matrices andprovide examples which shows that thisstatements arenot
valid for arbitrary GTP operators. Some conclusions are given in Section 14.
2. Exterior powers of the space Rn
LetRn denoten-dimensionalEuclideanspace, and(Rn)′ denoteanadjoint
space of all linear functionals on Rn. Since (Rn)′ is also n-dimensional, we
consider linear functionals from (Rn)′ as vectors from Rn.
Let us recall some basic definitions and statements about the tensor and
4
exterior powers of the space Rn (for more complete information see [6, 14,
23]).
Let j = 2, ..., n. The space of all multilinear functionals on ×j(Rn)′
is called the jth tensor power of the space Rn and denoted by ⊗jRn. Its
elements are called tensors.
Let x , ..., x be arbitrary vectors from Rn. Then the multilinear func-
1 j
tional x ⊗...⊗x : (×j(Rn)′) → R which acts according to the rule
1 j
(x ⊗...⊗x )(f ,...,f ) = hx ,f i...hx ,f i,
1 j 1 j 1 1 j j
is called a tensor product of the vectors x , ..., x . (Here the linear func-
1 j
tionals f , ..., f ∈ (Rn)′ are considered as vectors from Rn).
1 j
The j-th tensor power ⊗jRn of the space Rn is spanned by elementary
tensor products of the form x ⊗...⊗x where x , ..., x ∈ Rn. Examine
1 j 1 j
an arbitrary basis e , ..., e in Rn. Then all the possible tensor products
1 n
of the form e ⊗...⊗ e (1 ≤ i , ..., i ≤ n) of the initial basic vectors
i1 ij 1 j
form a basis in ⊗jRn. It follows that the space ⊗jRn is finite-dimensional
with dim(⊗jRn) = nj.
Let (i , ..., i ) be a permutation of the set [j] = {1, ..., j}. Define
1 j
1, if the permutation (i , ..., i ) is even;
1 j
χ(i ,...,i ) =
1 j
( −1, if the permutation is odd.
The jth exterior power ∧jRn of the space Rn is a subspace of the space
⊗jRn consisting of all antisymmetric tensors (i.e. all the tensors ϕ for which
ϕ(f ,...,f ) = χ(i ,...,i )ϕ(f ,...,f ) where f ,...,f are arbitrary func-
1 j 1 j i1 ij 1 j
tionals from (Rn)′).
Let x ,..., x be arbitrary vectors from Rn. Then the multilinear func-
1 j
tional x ∧...∧x : ×j(Rn)′ → R which acts according to the rule
1 j
(x ∧...∧x )(f ,...,f ) = χ(i ,...,i )(x ⊗...⊗x )(f ,...,f ) =
1 j 1 j 1 j i1 ij 1 j
(i1X,...,ij)
= χ(i ,...,i ) hx ,f i...hx ,f i
1 j i1 1 ij j
(i1X,...,ij)
is called an exterior product of the vectors x , ..., x . Here the sum is taken
1 j
with respect to all the permutations (i , ..., i ) of [j] and linear functionals
1 j
f , ..., f ∈ (Rn)′ are considered as vectors from Rn.
1 j
5
It is easy to see, that the exterior product x ∧...∧x is antisymmetric,
1 j
i.e. the following equality holds for every permutation (i , ..., i ) of [j]:
1 j
x ∧...∧x = χ(i , ...,i )(x ∧...∧x ).
i1 ij 1 j 1 j
The space ∧jRn is spanned by all the exterior products x ∧ ... ∧ x
1 j
where x , ..., x ∈ Rn. If the vectors e , ..., e form a basis in the initial
1 j 1 n
space Rn then the set of all exterior products of the type {e ∧ ... ∧ e }
i1 ij
where 1 ≤ i < ... < i ≤ n forms a canonical basis in the space ∧jRn (see
1 j
[5, 15, 17]). Thus thespace ∧jRn is finite dimensional with dim(∧jRn) = n .
j
(Here n = n! ).
j j!(n−j)! (cid:0) (cid:1)
A scalar product on ∧jRn is defined by the formula:
(cid:0) (cid:1)
hx ∧...∧x , y ∧...∧y i = (x ∧...∧x )(y ,...,y ) =
1 j 1 j 1 j 1 j
= χ(i ,...,i ) hx ,y i...hx ,y i.
1 j i1 1 ij j
(i1X,...,ij)
It follows that the adjoint space (∧jRn)′ can be considered as ∧j(Rn)′ (see
[23], p. 88).
Let the element ϕ ∈ ∧jRn be represented in the form of the exterior
product x ∧...∧x of some vectors x , ..., x ∈ Rn. Then ϕ is called a
1 j 1 j
simple j-vector. The set of all simple j-vectors is called the Grassmann cone
and denoted ⊼jRn. The equality ⊼jRn = ∧jRn holds only for j = 1,n − 1
and n. (Note that ∧1Rn = Rn and ∧nRn = R). If j = 2, ..., n−2, then we
can find elements of ∧jRn which can not be represented as simple j-vectors
(see [17], p. 83). It is not difficult to see that the set ⊼jRn is uniform (i.e.
the equality α⊼j Rn = ⊼jRn is true for every nonzero α ∈ R) and closed in
the space ∧jRn.
Let us define a map A acting from the set of all j-dimensional subspaces
j
of Rn to the set of 1-dimensional subspaces (i.e. lines) of ∧jRn according to
the following rule:
Aj(L) = {t(x1 ∧...∧xj)}t∈R,
where Lis aj-dimensional subspace ofRn, x , ..., x arej arbitrarylinearly
1 j
independent vectors from L.
It is not difficult to see that the map A is well-defined, i.e. if x , ..., x
j 1 j
and y , ..., y are two sets of linearly independent vectors, which belongs to
1 j
the same j-dimensional subspace L, then their exterior products x ∧...∧x
1 j
and y ∧...∧y are collinear (see, e.g., [17]).
1 j
6
The map A is a bijective map between all j-dimensional subspaces of Rn
j
and all lines of ⊼jRn (see [17], p. 86).
Let us consider the (n−1)th exterior power of the n-dimensional space
Rn. Note that n = n, thus dim(∧n−1Rn) = n. All the exterior products
n−1
of the type {e ∧ ... ∧ e } where 1 ≤ i < ... < i ≤ n of the initial
i1(cid:0) (cid:1) in−1 1 n−1
basic vectors e , ..., e form a basis in ∧n−1Rn. Let us define a bijective
1 n
linear operator J : ∧n−1Rn → Rn in the following way:
n
J (e ∧...∧e ) = (−1)k+1e ,
n i1 in−1 k
where k = [n]\{i , ..., i }.
1 n−1
Let x = (x1, ..., xn) where i = 1, ..., n−1 be n−1 arbitrary linearly
i i i
independent vectors from Rn. Write the exterior product x ∧...∧x in
1 n−1
the form
xi1 ... xin−1
1 1
x ∧...∧x = ... ... ... (e ∧...∧e ).
1 n−1 (cid:12) (cid:12) i1 in−1
(i1,X...,in−1)(cid:12)(cid:12)xin1−1 ... xinn−−11(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
It is not difficult to see, that t(cid:12)he vector (cid:12)
e ... e
n xi1 ... xin−1 1 n
1 1 x1 ... xn
J(x ∧...∧x ) = ... ... ... (−1)k+1e = (cid:12) 1 1 (cid:12)
1 n−1 (cid:12) (cid:12) k (cid:12) ... ... ... (cid:12)
Xk=1(cid:12)(cid:12)xin1−1 ... xinn−−11(cid:12)(cid:12) (cid:12)(cid:12)x1 ... xn (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) n−1 n−1(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
is orthogonal to the hyperplane spanned by the vectors x , ..., x .
1(cid:12) n−1 (cid:12)
3. Conic sets: basic definitions and statements
Let us recall some basic definitions of the theory of cones (see [3, 11, 21,
22]).
A closed subset K ⊂ Rn is called a proper cone, if it is a convex cone (i.e.
for any x,y ∈ K, α ≥ 0 we have x+y, αx ∈ K), pointed (K∩(−K) = {0})
and solid (int(K) 6= ∅).
The set K∗ ⊂ (Rn)′ defined in the following way
K∗ = {x∗ ∈ (Rn)′ : ∀y ∈ K hy,x∗i ≥ 0},
7
is called the adjoint cone to the cone K. The set K is a proper cone in Rn
if and only if K∗ is a proper cone in (Rn)′. The interior of K∗ is defined by
the equality
int(K∗) = {x∗ ∈ (Rn)′ : ∀y ∈ K hy,x∗i > 0}.
Example 1. Let x , ..., x ∈ Rn be linearly independent vectors. The
1 n
set
n
K = { c x : c , ..., c ≥ 0}
i i 1 n
i=1
X
of all linear combinations of the vectors x , ..., x with nonnegative coeffi-
1 n
cients is a proper cone in Rn. Such a cone is called spanned by the vectors
x , ..., x . The cone spanned by the basic vectors e , ..., e is denoted
1 n 1 n
by Rn.
+
Example 2. The set
K = {x = (x1, ..., xn) ∈ Rn;
(x1)2 +...+(xk−1)2 +(xk+1)2 +...+(xn)2 ≤ xk}
is a proper cpone in Rn. Such a cone is called an ice-cream cone.
Example 3. Let F ⊂ Rn be a closed, convex and bounded set. The set
K(F) of all elements of the form αx where α ≥ 0, x ∈ F is a pointed convex
cone in Rn. If int(F) 6= ∅, then the cone K(F) is obviously proper.
Let us examine the cones spanned by the vectors ǫ e , ..., ǫ e where
1 1 n n
each ǫ (i = 1, ..., n) is equal to +1 or −1. This cone is called a basic cone.
i
The space Rn with a fixed basis e , ..., e consists of 2n basic cones, one of
1 n
which is Rn.
+
We list some properties of basic cones which will be used later.
1. The projection of any basic cone on any basic subspace (i.e on a
subspace spanned by any subsystem of the initial basic vectors) is a basic
cone in this subspace.
2. If K is a basic cone in Rn, then the adjoint cone K∗ is also a basic
cone in (Rn)′.
As it was mentioned above, every basis e , ..., e in Rn defines a basis
1 n
in the space ∧jRn = n which consists of all exterior products of the form
j
{e ∧...∧e }, where 1 ≤ i < ... < i ≤ n. Denote the cone spanned by
i1 ij (cid:0) (cid:1) 1 j
this exterior basic vectors by ∧jRn. Let us call a cone in ∧jRn spanned by
+
the simple j-vectors of the form ±(e ∧...∧e ) where 1 ≤ i < ... < i ≤ n
i1 ij 1 j
8
an exterior basic cone defined by the basis e , ..., e . It is easy to see, that
1 n
not every basic cone in ∧jRn is an exterior basic cone.
We list some obvious properties of exterior basic cones.
1. Let Lbe any basic subspace ofRn. Then the projection of any exterior
basic cone on the subspace ∧jL of the space ∧jRn is an exterior basic cone
in this subspace.
2. If K is an exterior basic cone in ∧jRn, then the adjoint cone K∗ is an
j j
exterior basic cone in (∧jRn)′ = ∧j(Rn)′.
Let us recall the following characterization of a proper cone K (see, for
example, [7]).
The angle θ (K) defined by the equality
max
θ (K) = sup arccoshx,yi,
max
x,y∈K∩Sn
where S is the unit sphere in Rn, is called the maximal angle of the cone K.
n
Any basic cone in Rn can be converted using some linear transformation
to the cone Rn. Thus we can assume without loss of generality that any
+
π
basic cone K satisfies the inequality θ (K) ≤ .
max
2
Besides cones we shall be interested in some other sets in Rn. Recall the
definitions of the following conic sets (see [11]).
A closed subset T ⊂ Rn is called a cone of rank k (0 ≤ k ≤ n) if for every
x ∈ T, α ∈ R the element αx ∈ T and there is at least one k-dimensional
subspace and no higher dimensional subspaces in T.
For the definition and examples of cones of rank k see also [12, 20, 21].
Note that a cone of rank k is usually not convex.
Example 1. Let L , ..., L be subspaces of Rn with maxdim(L ) = k.
1 m i
i
m
Then L is a cone of rank k in Rn.
i
i=1
ExSample 2. Let K ⊂ Rn be a proper cone. Then K ∪(−K) is a cone
of rank 1 in Rn, and Rn \(int(K)∪int(−K)) is a cone of rank n−1 in Rn.
T K
4. Set ( ) and its properties
j
Given a proper cone Kj ⊂ ∧jRn = R(nj), j = 2, ..., n. Let us define the
set T(K ) ⊂ Rn in the following way:
j
T(K ) = {x ∈ Rn : ∃ x , ..., x ∈ Rn, for which
j 1 2 j
9
x ∧x ∧...∧x ∈ (int(K )∪int(−K ))}.
1 2 j j j
Let us define the set T(K ) in the following way:
j
T(K ) = {xb∈ Rn : ∃ x , ..., x ∈ Rn, for which
j 1 2 j
x ∧x ∧...∧x ∈ (K ∪(−K ))\{0}}∪{0}.
1 2 j j j
b
It is not difficult to see, that the sets T(K ) and T(K ) may not coincide
j j
for an arbitrary proper cone K ⊂ ∧jRn.
j
The followinglemma describes thestructure ofthebsets T(K ) andT(K ).
j j
Lemma 3. Let K ⊂ ∧jRn be a proper cone. Then the set T(K ), if itbis not
j j
{0}, coincides with the set of all j-dimensional subspaces L ⊂ Rn for which
corresponding lines A (L) belong to K ∪(−K ). The set Tb(K ), if it is not
j j j j
empty, coincides with the closure of the set of all j-dimensional subspaces
L ⊂ Rn for which corresponding lines A (L) belong to int(K )∪int(−K ).
j j j
Proof. ⇐ The inclusion 0 ∈ T(K ) follows from the definition of the set
j
T(K ). Letanarbitrarynonzerovectorx belongtoaj-dimensionalsubspace
j 1
Lfor which the corresponding linbe A (L) belongs to K ∪(−K ). Let us show
j j j
tbhat x ∈ T(K ). Indeed, let us find vectors x , ..., x such that the system
1 j 2 j
{x , x , ..., x } forms a basis of the j-dimensional subspace L. Examine
1 2 j
the exteriobr product x ∧ x ∧ ... ∧ x . Since x , x , ..., x are linearly
1 2 j 1 2 j
independent, the element x ∧x ∧...∧x is nonzero and belongs to the line
1 2 j
A (L) ⊂ (K ∪ (−K )). Since x ∧ x ∧ ... ∧ x ∈ (K ∪ (−K )) \ {0} for
j j j 1 2 j j j
some nonzero vectors x , ..., x ∈ Rn, we have x ∈ T(K ).
2 j 1 j
⇒ The inclusion 0 ∈ L is obvious for any subspace L ⊂ Rn. Let x ∈
1
T(K ) be nonzero. Then there exist nonzero vectorsbx , ..., x ∈ Rn for
j 2 j
which x ∧x ∧...∧x ∈ (K ∪(−K ))\{0}. Since x ∧x ∧...∧x 6= 0,
1 2 j j j 1 2 j
tbhey are linearly independent. Examine the j-dimensional subspace L =
Lin(x , x , ..., x ). Since K ∪ (−K ) is a cone of rank 1 in ∧jRn, the
1 2 j j j
line {t(x1 ∧ x2 ∧ ... ∧ xj)}t∈R corresponding to the subspace L belongs to
K ∪(−K ).
j j
The second part of the lemma is proved analogically.
Now examine the set T(K ) defined in the following way:
j
T(K ) = {x e∈ Rn : ∃ x , ..., x ∈ Rn, for which
j 1 2 j
e 10