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On sufficient conditions for the total positivity of matrices
Introduction
Definition
A real matrix Ais said to be k-times positive, if all minors of Aof
order not greater than kare non-negative. A matrix Ais said to
be totally positive, if all minors of Aare non-negative.
According to S. Karlin we will denote the class of all k-times
positive matrices by TPkand the class of all totally positive
matrices by TP.By STP we will denote the class of matrices
with all minors being strictly positive and by STPkthe class of
matrices with all minors of order not greater than kbeing strictly
positive.
On sufficient conditions for the total positivity of matrices
Introduction
Definition
A real matrix Ais said to be k-times positive, if all minors of Aof
order not greater than kare non-negative. A matrix Ais said to
be totally positive, if all minors of Aare non-negative.
According to S. Karlin we will denote the class of all k-times
positive matrices by TPkand the class of all totally positive
matrices by TP.By STP we will denote the class of matrices
with all minors being strictly positive and by STPkthe class of
matrices with all minors of order not greater than kbeing strictly
positive.
On sufficient conditions for the total positivity of matrices
Consider a minor of order 2 of a matrix A= (ai,j)formed by
consecutive rows and columns:
ai,jai,j+1
ai+1,jai+1,j+1
.This minor
is positive if aij ai+1,j+1>ai,j+1ai+1,j.
We will see that if a matrix entries satisfy a stronger inequality
then all minors of a matrix are nonnegative.
On sufficient conditions for the total positivity of matrices
T. Craven and G. Csordas (1998) proved the following theorem
(very useful and easily verified sufficient condition for strict total
positivity of a matrix)
Theorem A. Denote by ˜
c the unique real root of
x3−5x2+4x−1=0(˜
c≈4.0796). Let M = (aij )be an n ×n
matrix with the property that
(a) aij >0(1≤i,j≤n)and
(b) aij ai+1,j+1≥˜
c ai,j+1ai+1,j(1≤i,j≤n−1).
Then M is strictly totally positive.
Note that the verification of total positivity is, in general, a very
difficult problem. Theorem A provides a convenient sufficient
condition for total positivity of a matrix.
On sufficient conditions for the total positivity of matrices
T. Craven and G. Csordas (1998) proved the following theorem
(very useful and easily verified sufficient condition for strict total
positivity of a matrix)
Theorem A. Denote by ˜
c the unique real root of
x3−5x2+4x−1=0(˜
c≈4.0796). Let M = (aij )be an n ×n
matrix with the property that
(a) aij >0(1≤i,j≤n)and
(b) aij ai+1,j+1≥˜
c ai,j+1ai+1,j(1≤i,j≤n−1).
Then M is strictly totally positive.
Note that the verification of total positivity is, in general, a very
difficult problem. Theorem A provides a convenient sufficient
condition for total positivity of a matrix.
On sufficient conditions for the total positivity of matrices
We will denote by TP2(c)the class of all matrices
M= (aij ),1≤i≤m,1≤j≤n,(m,n∈N∪ ∞ ) with positive
entries which satisfy the condition
aij ai+1,j+1≥c ai,j+1ai+1,j(1≤i<m,1≤j<n).(1)
We will denote by STP2(c)the class of all matrices
M= (aij ),1≤i≤m,1≤j≤n,(m,n∈N∪ ∞ ) with positive
entries which satisfy the condition
aij ai+1,j+1>c ai,j+1ai+1,j(1≤i<m,1≤j<n).(2)
It is easy to see that TP2=TP2(1)and STP2=STP2(1).
Theorem A states that TP2(˜
c)∩STP1⊂STP.
On sufficient conditions for the total positivity of matrices
We will denote by TP2(c)the class of all matrices
M= (aij ),1≤i≤m,1≤j≤n,(m,n∈N∪ ∞ ) with positive
entries which satisfy the condition
aij ai+1,j+1≥c ai,j+1ai+1,j(1≤i<m,1≤j<n).(1)
We will denote by STP2(c)the class of all matrices
M= (aij ),1≤i≤m,1≤j≤n,(m,n∈N∪ ∞ ) with positive
entries which satisfy the condition
aij ai+1,j+1>c ai,j+1ai+1,j(1≤i<m,1≤j<n).(2)
It is easy to see that TP2=TP2(1)and STP2=STP2(1).
Theorem A states that TP2(˜
c)∩STP1⊂STP.
On sufficient conditions for the total positivity of matrices
We will denote by TP2(c)the class of all matrices
M= (aij ),1≤i≤m,1≤j≤n,(m,n∈N∪ ∞ ) with positive
entries which satisfy the condition
aij ai+1,j+1≥c ai,j+1ai+1,j(1≤i<m,1≤j<n).(1)
We will denote by STP2(c)the class of all matrices
M= (aij ),1≤i≤m,1≤j≤n,(m,n∈N∪ ∞ ) with positive
entries which satisfy the condition
aij ai+1,j+1>c ai,j+1ai+1,j(1≤i<m,1≤j<n).(2)
It is easy to see that TP2=TP2(1)and STP2=STP2(1).
Theorem A states that TP2(˜
c)∩STP1⊂STP.
On sufficient conditions for the total positivity of matrices
The main result
Theorem 1 (Olga Katkova and A.V.). Denote by
ck:= 4 cos2π
k+1,k=2,3,4, . . . . For every c ≥ckwe have
(i) if M ∈TP2(c)then M ∈TPk;
(ii) if M ∈STP2(c)then M ∈STPk.
The following fact is a simple consequence of this theorem.
Theorem 2 (Olga Katkova and A.V.). For every c ≥4we have
(i) if M ∈TP2(c)then M ∈TP;
(ii) if M ∈STP2(c)then M ∈STP.
On sufficient conditions for the total positivity of matrices
The main result
Theorem 1 (Olga Katkova and A.V.). Denote by
ck:= 4 cos2π
k+1,k=2,3,4, . . . . For every c ≥ckwe have
(i) if M ∈TP2(c)then M ∈TPk;
(ii) if M ∈STP2(c)then M ∈STPk.
The following fact is a simple consequence of this theorem.
Theorem 2 (Olga Katkova and A.V.). For every c ≥4we have
(i) if M ∈TP2(c)then M ∈TP;
(ii) if M ∈STP2(c)then M ∈STP.
On sufficient conditions for the total positivity of matrices
The constants in theorems 1 and 2 are sharp not only in the
class of matrices with nonnegative entries but in the classes of
Toeplitz matrices and of Hankel matrices.
(A matrix Mis a Toeplitz matrix if M= (aj−i)and a matrix Kis a
Hankel matrix if K= (bj+i).
Theorem 3 (Olga Katkova and A.V.).
(i) For every 1≤c<ckthere exists a k ×k Toeplitz matrix
M∈TP2(c)with det M<0;
(ii) for every 1≤c<ckthere exists a k ×k Hankel matrix
K∈TP2(c)with det K<0.
On sufficient conditions for the total positivity of matrices
The constants in theorems 1 and 2 are sharp not only in the
class of matrices with nonnegative entries but in the classes of
Toeplitz matrices and of Hankel matrices.
(A matrix Mis a Toeplitz matrix if M= (aj−i)and a matrix Kis a
Hankel matrix if K= (bj+i).
Theorem 3 (Olga Katkova and A.V.).
(i) For every 1≤c<ckthere exists a k ×k Toeplitz matrix
M∈TP2(c)with det M<0;
(ii) for every 1≤c<ckthere exists a k ×k Hankel matrix
K∈TP2(c)with det K<0.
On sufficient conditions for the total positivity of matrices
The constants in theorems 1 and 2 are sharp not only in the
class of matrices with nonnegative entries but in the classes of
Toeplitz matrices and of Hankel matrices.
(A matrix Mis a Toeplitz matrix if M= (aj−i)and a matrix Kis a
Hankel matrix if K= (bj+i).
Theorem 3 (Olga Katkova and A.V.).
(i) For every 1≤c<ckthere exists a k ×k Toeplitz matrix
M∈TP2(c)with det M<0;
(ii) for every 1≤c<ckthere exists a k ×k Hankel matrix
K∈TP2(c)with det K<0.
On sufficient conditions for the total positivity of matrices
Corollary of Theorem 3.
(i) For every 1≤c<4there exists a Toeplitz matrix
M∈TP2(c)such that M /∈TP;
(ii) for every 1≤c<4there exists a Hankel matrix K ∈TP2(c)
such that M /∈TP.
A variation of theorem 2 (i) for the class of Toeplitz matrices
was proved by J. I. Hutchinson in 1923.
On sufficient conditions for the total positivity of matrices
Corollary of Theorem 3.
(i) For every 1≤c<4there exists a Toeplitz matrix
M∈TP2(c)such that M /∈TP;
(ii) for every 1≤c<4there exists a Hankel matrix K ∈TP2(c)
such that M /∈TP.
A variation of theorem 2 (i) for the class of Toeplitz matrices
was proved by J. I. Hutchinson in 1923.
On sufficient conditions for the total positivity of matrices
m-times positive sequences
The class of m-times positive sequences consists of the
sequences (ak)∞
k=0such that all minors of the infinite matrix
a0a1a2a3. . .
0a0a1a2. . .
0 0 a0a1. . .
000a0. . .
.
.
..
.
..
.
..
.
....
(3)
of order not greater than mare non-negative. The class of
m-times positive sequences is denoted by PFm.A sequence
(ak)∞
k=0is called totally positive sequence if all minors of the
infinite matrix (3) are nonnegative. The class of totally positive
sequences is denoted by PF∞.
On sufficient conditions for the total positivity of matrices
The classical result by Aissen, Schoenberg,
Whitney and Edrei
The multiply positive sequences (also called Pólya frequency
sequences) were introduced by Fekete in 1912 (see [7]) in
connection with the problem of exact calculation of the number
of positive zeros of a real polynomial.
The class PF∞was completely described by Aissen,
Schoenberg, Whitney and Edrei in [1] (see also [13], p. 412):
Theorem ASWE (1952). A function f ∈PF∞iff
f(z) = Czneγz
∞
Y
k=1
(1+αkz)/(1−βkz),
where C ≥0,n∈Z, γ ≥0, αk≥0, βk≥0,P(αk+βk)<∞.
On sufficient conditions for the total positivity of matrices
The classical result by Aissen, Schoenberg,
Whitney and Edrei
The multiply positive sequences (also called Pólya frequency
sequences) were introduced by Fekete in 1912 (see [7]) in
connection with the problem of exact calculation of the number
of positive zeros of a real polynomial.
The class PF∞was completely described by Aissen,
Schoenberg, Whitney and Edrei in [1] (see also [13], p. 412):
Theorem ASWE (1952). A function f ∈PF∞iff
f(z) = Czneγz
∞
Y
k=1
(1+αkz)/(1−βkz),
where C ≥0,n∈Z, γ ≥0, αk≥0, βk≥0,P(αk+βk)<∞.
On sufficient conditions for the total positivity of matrices
Hyperbolic polynomials
By theorem ASWE a polynomial p(z) = Pn
k=0akzk,ak≥0,
has only real zeros (i.e. is hyperbolic) if and only if the
sequence (a0,a1,...,an,0,0, . . .)∈PF∞.
In 1926, Hutchinson extended the work of Petrovitch (1908)
and Hardy (1904) and proved the following theorem.
Theorem B. Let f (z) = P∞
k=0akzk, ak>0,∀k.Inequalities
a2
n≥4an−1an+1,∀n≥1,holds if and only if the following two
properties hold:
(i) The zeros of f(x) are all real, simple and negative and
(ii) the zeros of any polynomial Pn
k=makzk, formed by taking
any number of consecutive terms of f(x), are all real and
non-positive.
On sufficient conditions for the total positivity of matrices
Hyperbolic polynomials
By theorem ASWE a polynomial p(z) = Pn
k=0akzk,ak≥0,
has only real zeros (i.e. is hyperbolic) if and only if the
sequence (a0,a1,...,an,0,0, . . .)∈PF∞.
In 1926, Hutchinson extended the work of Petrovitch (1908)
and Hardy (1904) and proved the following theorem.
Theorem B. Let f (z) = P∞
k=0akzk, ak>0,∀k.Inequalities
a2
n≥4an−1an+1,∀n≥1,holds if and only if the following two
properties hold:
(i) The zeros of f(x) are all real, simple and negative and
(ii) the zeros of any polynomial Pn
k=makzk, formed by taking
any number of consecutive terms of f(x), are all real and
non-positive.
On sufficient conditions for the total positivity of matrices
Hyperbolic polynomials
Theorem C (O.Holtz and M.Tyaglov, 2012) Let
P(x) = a0+a1x+a2x2+. . . +anxn,a06=0,an>0be a real
polynomial. Polynomial P has all real negative zeros if and only
if the infinite matrix
a0a1a2a3a4a5a6. . .
0a12a23a34a45a56a6. . .
0a0a1a2a3a4a5. . .
0 0 a12a23a34a45a5. . .
0 0 a0a1a2a3a4. . .
0 0 0 a12a23a34a4. . .
.
.
..
.
..
.
..
.
..
.
..
.
..
.
....
(5)
is totally positive.
On sufficient conditions for the total positivity of matrices
Stable polynomials
Some words about Hurwitz stable polyniomials.
Definition. A real polynomial P is called Hurwitz (stable) if all
its zeros have negative real parts, i.e. P (z0) = 0⇒Re z0<0.
Polynomial stability problems of various types arise in a number
of problems in mathematics and engineering.
On sufficient conditions for the total positivity of matrices
The following statement (usually attributed to A. Stodola) is the
well-known necessary condition for a real polynomial to be
stable.
Statement. P(z) = a0+a1z+. . . +anzn∈R[z],an>0,is
stable ⇒aj>0,0≤j≤n−1.
The Routh-Hurwitz Criterion .The polynomial
P(z) = a0+a1z+. . . +anzn,an>0,is stable if and only if
the first n principal minors of the corresponding Hurwitz matrix
H(F) :=
an−1an−3an−5. . . 0
anan−2an−4. . . 0
0an−1an−3. . . 0
0anan−2. . . 0
.
.
..
.
..
.
..
.
....
are positive.
On sufficient conditions for the total positivity of matrices
Using Theorem A and continuity reasonings D.K. Dimitrov and
J.M. Peña proved the following theorem.
Theorem D .Let ˜
c be defined as in Theorem A. If the
coefficients of P(z) = a0+a1z+. . . +anznare positive and
satisfy the inequalities
akak+1≥˜
cak−1ak+2for k =1,2,...,n−2,
then P(z)is Hurwitz. In particular, the conclusion is true if
a2
k≥√˜
cak−1ak+1for k =1,2,...,n−1.
Using theorem 1 one can improve the above theorem by
replacing the constant ˜
cby 4 cos2π
n+1.
On sufficient conditions for the total positivity of matrices
The natural problem is: for a given n∈Nwhat is the smallest
possible constant dnsuch that if the coefficients of
P(z) = a0+a1z+. . . +anznare positive and satisfy the
inequalities akak+1>dnak−1ak+2for k=1,2,...,n−2,
then P(z)is Hurwitz?
It is well-known (and easy verified) that polynomials of degree 1
and 2 with positive leading coefficient are stable if and only if all
their coefficients are positive. Polynomial P(z) = P3
j=0ajzjwith
positive coefficients is stable if and only if a1a2>a0a3(see, for
example, [25, p. 34]).
On sufficient conditions for the total positivity of matrices
Theorem 4 (Olga Katkova and A.V.). Let x0be the (unique)
positive root of the polynomial x3−x2−2x−1(x0≈2.1479).
1. If the coefficients of P (z) = P4
k=0akzkare positive and
satisfy the inequalities akak+1>2ak−1ak+2for k =1,2,
then P(z)is Hurwitz. In particular, the conclusion is true if
a2
k>√2ak−1ak+1for k =1,2,3.
2. If the coefficients of P (z) = P5
k=0akzkare positive and
satisfy the inequalities akak+1>x0ak−1ak+2for k =1,2,3,
then P(z)is Hurwitz. In particular, the conclusion is true if
a2
k>√x0ak−1ak+1for k =1,2,3,4.
3. If the coefficients of P (z) = Pn
k=0akzk,n>5,are positive
and satisfy the inequalities
akak+1≥x0ak−1ak+2for k =1,2,...,n−2,then P(z)is
Hurwitz. In particular, the conclusion is true if
a2
k≥√x0ak−1ak+1for k =1,2,...,n−1.
On sufficient conditions for the total positivity of matrices
Note that
akak+1
ak−1ak+2=a2
k
ak−1ak+1
a2
k+1
akak+2,
and thus the following theorem demonstrates that the constants
in Theorem 4 are the smallest possible for every n.
Theorem 5 (Olga Katkova and A.V.).
1. For every d ≤√2there exists a polynomial
P(z) = P4
k=0akzkwith positive coefficients under condition
a2
k=dak−1ak+1for k =1,2,3,such that P(z)is not Hurwitz.
2. For every d ≤√x0there exists a polynomial
P(z) = P5
k=0akzkwith positive coefficients under condition
a2
k=dak−1ak+1for k =1,2,3,4,such that P(z)is not
Hurwitz.
3. For every n >5and every ε > 0there exists a polynomial
P(z) = Pn
k=0akzkwith positive coefficients under condition
a2
k>(√x0−ε)ak−1ak+1for k =1,2,...,n−1,such that P(z)
is not Hurwitz.
On sufficient conditions for the total positivity of matrices
Theorem 4 may be generalized for entire functions as follows.
Theorem 6 (Olga Katkova and A.V.). If the coefficients of
G(z) = P∞
k=0akzkare positive and satisfy the inequalities
akak+1≥x0ak−1ak+2for k ∈N,then all zeros of G(z)have
negative real parts. In particular, the conclusion is true if
a2
k≥√x0ak−1ak+1for k ∈N.
On sufficient conditions for the total positivity of matrices
Positive polynomials
Positive polynomials arise in many important branches of
mathematics. We give a simple sufficient condition for an even
degree polynomial with positive coefficients to be positive on
the real line. To prove this condition we apply Theorem 1.
Theorem 7 (Olga Katkova and A.V.). Let P(x) = P2n
k=0akxkbe
a polynomial with positive coefficients. If the inequalities
a2
2k+1
a2ka2k+2<1
cos2(π
n+2)
hold for all k =0,1,...,n−1,then P(x)>0for every x ∈R.
On sufficient conditions for the total positivity of matrices
Positive polynomials
Positive polynomials arise in many important branches of
mathematics. We give a simple sufficient condition for an even
degree polynomial with positive coefficients to be positive on
the real line. To prove this condition we apply Theorem 1.
Theorem 7 (Olga Katkova and A.V.). Let P(x) = P2n
k=0akxkbe
a polynomial with positive coefficients. If the inequalities
a2
2k+1
a2ka2k+2<1
cos2(π
n+2)
hold for all k =0,1,...,n−1,then P(x)>0for every x ∈R.
On sufficient conditions for the total positivity of matrices
The constants 1
cos2(π
n+2)in Theorem 4 is sharp for every n∈N.
Theorem 8. For every n ∈Nthere exists a polynomial
Q(x) = P2n
k=0akxkwith positive coefficients under conditions
a2
2k+1
a2ka2k+2=1
cos2(π
n+2),k=0,1,...,n−1,
and the polynomial Q(x)has not less than two real zeros.
On sufficient conditions for the total positivity of matrices
The following statement is a simple corollary of Theorem 7.
Corollary Let P(x) = P2n+1
k=0akxkbe a polynomial with positive
coefficients. If the inequalities
a2
2k
a2k−1a2k+1<4k2−1
4k2·1
cos2(π
n+2)
hold for all k =1,2,...,n,then P(x)has only one real zero
(counting multiplicities).
On sufficient conditions for the total positivity of matrices
We show that the constants in the last statement is also sharp
for every n∈N.
Theorem 9. For every n ∈Nthere exists a polynomial
Q(x) = P2n+1
k=0akxkwith positive coefficients under conditions
a2
2k
a2k−1a2k+1=4k2−1
4k2·1
cos2(π
n+2),k=1,2,...,n,
and the polynomial Q(x)has not less than three real zeros.
On sufficient conditions for the total positivity of matrices
We show that the constants in the last statement is also sharp
for every n∈N.
Theorem 9. For every n ∈Nthere exists a polynomial
Q(x) = P2n+1
k=0akxkwith positive coefficients under conditions
a2
2k
a2k−1a2k+1=4k2−1
4k2·1
cos2(π
n+2),k=1,2,...,n,
and the polynomial Q(x)has not less than three real zeros.
On sufficient conditions for the total positivity of matrices
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On sufficient conditions for the total positivity of matrices
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On sufficient conditions for the total positivity of matrices
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On sufficient conditions for the total positivity of matrices
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On sufficient conditions for the total positivity of matrices
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