Toeplitz matrices with totally nonnegative inverses

Toeplitz

Matrices

With Totally Nonnegative

Inverses

Jens Lorenz

hstitut fiir Numerische und instrumentelle der Universitiit Miinster Roxeler Strasse 64 D-4400 Miinster, Germany

Mathemutik

and Wolfgang Mackens Institut fiir Mathemutik der Ruhr-Universitiit Bochum Universitiitsstrasse 150 D-4630 Bochum, Germany

Submittedby Ky Fan

ABSTRACT It is shown that the inverse of a Toeplitz matrix has only nonnegative minors if the zeros of a certain polynomial are positive or if their arguments are less than r/( k + n), where n is the dimension and k + 1 is the bandwidth of the matrix.

1.

INTRODUCTION

A real n X n matrix A = (aii) is called inverse monotone [2] (or of monotone kind [5] or inverse positive [20] or monotone [22]) iff A is nonsingular and all elements of A - ’ are nonnegative. If aii < 0 for all i #j, and A is inverse monotone, then A is called an M-matrix [16]. The subclass of M-matrices has been intensively investigated, and several conditions are well known for A to be inverse monotone if aii < 0 for all i #j (see [17] and the references given there). In case that aii > 0

LINEAR ALGEBRA

for some

AND ITS APPLICATIONS

0 ELsevierNorth Holland,Inc., 1979

i # j,

24:133-141

(I)

(1979)

133

0024-3795/79/020133 + 9$01.75

134

JENS LORENZ AND WOLFGANG

MACKENS

question of inverse monotonicity is in general more involved. Some sufficient criteria for this case are described in [3, 4, 12, 13, 181, where applications to discrete analogs of boundary value problems are also discussed. Another class of inverse monotone matrices with the property (1) is considered in [14], where it is shown that symmetric quindiagonal Toeplitz matrices are inverse monotone if a corresponding polynomial has only real and positive zeros. In our paper we are going to generalize Meek’s result in three aspects: the

(1) General finite Toeplitz matrices are considered. (2) Certain complex zeros are allowed. (3) It is shown that not only the elements but also all minors of A -I are nonnegative. Toeplitz matrices play an important role in the theory of discrete random processes [21] and other branches of applied mathematics [l, Chapter 121. In addition, with a result of Schroder’s [ 191 the inverse monotonicity of Toeplitz matrices can be used in the study of inverse monotonicity of general band matrices. We give an application to a discretization of a fourth-order boundary value problem. 2.

THE MAIN RESULT

A real matrix B is called totally nonnegative (we write R ; 0) if all minors of B are nonnegative [7, 111. Here a minor of B = (bii) E IX”*” is a determinant of a submatrix

where l
1.


l

l
Let A be a real Toeplitz matrix of the form

a0

a,

a-,

.

.

.

.

A=

.*.

a,

0 a,

E ia”,“, a-1 a1

(2)

135

MATRICES WITH NONNEGATIVE INVERSES

with 1~ 1, r
and a, > 0. Assume that the polynomial

pk(z) = a,xk + a,_,zk-’

+ *. + + a_Cl_lJz + a-,

(3)

of degree k = 1+ r has zeros Zl = r,e@l ,...,zk such that 5 > 0 and 1~~1<~/(n A-’

= rke’*,

+ k) ( j = 1,. . . , k). Then A is nonsingular and

$ 0.

If B = (b,,) is any real matrix, let B* = (b$) be defined by b$ = ( - l)‘+fbii [7]. The following lemma summarizes some known results on totally nonnegative matrices [7, Chapter 21. LEMMA1. (i)

If B

is nonsingular, then B _ ’ $ 0 iff B* ; 0.

(ii) If B is a tridiagonal M-matrix, then B - ’ $ 0. (iii) Any product of totally nonnegative matrices is totally nonnegative. The following lemma will be used in our proof of Theorem 1 to treat complex zeros. LEMMA2.

Let D E R”~” be the real matrix

... b

0

a

Assume re ‘iv are the zeros of ,?+az+b, where r>O and O
Then D-’

; 0.

1

JENS LORENZ AND WOLFGANG

136

MACKENS

We have D = PQ, where

Proof.

11

0

P2

...

.. .. .

P=

I

L0 pi=

-r

0

1

92

Q=

0

Pm1

.*. . *

.

.

q:

sin i’p qi = -rsin(j-1)cp

sin( j-2)(p sin(j-1)cp’

. 1

(j=2,...,m).

Since pi < 0, qi < 0, the matrices P and Q are M-matrices. Lemma 1 yields D-’

n

> 0.

Proof of Theorem 1. 1. The given matrix A is a submatrix of the following rn X m subdiagonal matrix D. Here m = n + k.

. . D=

.

. ’

a-1 .

r

, a,

.-.

7

a,

I

I. .

I

I .

I

’ a-,

4

I

I . 1

. .

I. .

.I.

Assume pk(z) = uJ,(t - r,)&(z_” + a+ + b,), where the zeros of z’+ uiz + bj are nonreal. Then D= uJiDifliDj, where Di,DiEW’~“’ are Toeplitz matrices corresponding to the factors of pk(z). Di is bidiagonal, 1 is in the main diagonal, - rj is in the lower secondary diagonal. The matrices 4 are of the form (4). By Lemma l(ii) we have Die’ $ 0, and Lemma 2 yields 4-l Thus (sgnu,)D-’

; 0,

(sgnu,)D*

; 0.

; 0.

MATRICES WITH NONNEGATIVE

By Obreschkoffs

theorem [15, Satz 17.31 we have sgnai

and rtherefore

137

INVERSES

=

(j= -E,...,r-I),

-Waj+l

sgna, = ( - 1)‘. Now A*

is a submatrix of ( - l)D*;

thus

A* > 0. Our assertion is proved by Lemma l(i) if we finally show that A is nonsingular. 2. Let R denote the n X (n + k) matrix consisting of the last n rows of D. Furthermore, let pi,. . . , pp be the distinct real (positive!) zeros of pk(x) with multiplicities m,, . , . , mp, and let r,e Zi’rl,. . . , rye ‘*p be the distinct nonreal zeros with multiplicities ni, . . . , n,,. Define the functions

u$$(t) = PP,‘,

R

tE

zQi( t) = tPT,sincp,t,

(p=o

)...)

m,-1,

r=l,...,

/4),

(p=o )...) nr-1,

:‘,“,

z@(t) = tv 7 cosp, 7t T

7=1,...,

V),

1

and the corresponding vectors

These k vectors nJ$ form a basis of the m&pace kerR = { y~!R’“:Ry=0} of R [lo, Chapter 51. Now assume x E I? and Ax =O. If y: =

(~o,.;.,o,,x,x”,,o,.;.,o) E R”, )...)

1

r

then Ry = Ax = 0,and therefore y is a linear combination of the vectors @,. The corresponding linear combination w(t) of the functions u$ vanishes at the k points t = 1,2,...,1

and

t=n+Z+l,n+Z+2

,..., m.

With the methods used in [ll, Chapter 61 it is easily seen that the k functions u$, form a Tchebyscheff system on the interval [l,m]. Hence the linear combination w(t) vanishes identically, which implies y = 0, x = 0. n An equivalent result to Theorem 1 is

138

JENS LORENZ AND WOLFGANG

THEOREM 2. Let A be given as in (2) with 1 0, and let pk(z) be the polynomial (3). If the zeros of

MACKENS

a_l#O#ar,

d4 = Pk4 satisfy the conditions of Theorem 1, then A is nonsingular and A >: 0. Proof.

The matrix A: satisfies the conditions of Theorem 1. Thus At* is

nonsingular and (A*)-’

3.

APPLICATIONS

> 0. We apply Lemma l(i), and find A =A**

TO GENERAL

> 0. n

BAND MATRICES

If we combine our Theorem 1 with a result of Schriider’s, we can prove inverse monotonicity for band matrices which are not necessarily of Toeplitz form. The following notation is used:

Here A=(u,[), R”.

A>B

H

X>Y X>Y B=(bii)

aii >bii

for all i, i,

ej

xi > yi

for all i,

@

xi > yi

for all i.

are real matrices and x = (xi), y = ( yi) are vectors in

THEOREM3 [19]. Let A,B EIIF”, B >A. IfB is inverse monotone, then A is inverse monotone if and only if there exists e E R” such that e>O

EXAMPLE.

and

Ae>O.

Consider the boundary-value problem

XIV(S) + q(s)+) x(0)=x”(O)

=f(s), + ax’(O) =x(l)

=x”(l)

- #8x’(l) =o.

(Bw)

Take a step size h = l/(n + 1) (n E IV), and use central difference fomulas of

MATRICES WITH NONNEGATIVE

139

INVERSES

order 2. Then the discretization matrix A,, E IV*” reads 2-ah A, = T + diag - -2+ahh

_4

6

+q1$qv.)...,qn-1, -4

2-fib -mh

-4

+qn

0

1

-4 T=h-4

1

1

3

-4 1

-4

6_

where qi = q(ih) and diag(a,, . . . , a,) is the n X n diagonal matrix with entries ai,.*., cu,. Assume that 0
G 9i

qmin

<

(i=l,...,n).

(Imax

Then the estimate To + qminZ< A,, < T + q_Z If e : = T, = T - diag(he4, 0,. . . ,O, h-4). where holds, (sinrh,sin2rh,. . . , sinnrh), then Toe=&,e with h,,: =h-4(2-2cosrh)2a 7~~;thus A,, e > 0 as long as qmin> -A,,. An application of Theorem 1 requires knowledge of the zeros of the polynomial (3):

h4P4b)

=

z4

-

4z3

+

(6+ h4q_)z2

- 4,~ + 1 = (z - 1)4 + h4q_z2.

As elementary discussion shows that the polynomial

(2 - 1)4 + 4C2Z2

WO)

has the zeros

if QE(O,a/2)

is the solution of sincptang, = c

JENS LORENZ AND WOLFGANG

140

MACKENS

and

Without loss of generality we can assume qmax>O, and we find with Theorem 1 that the matrix T + q ,,I has a totally nonnegative inverse if

71

sin -tan n+4

-

GE

71

n+4

’

2(n+1)2’

Thus by Theorem 3 we get the following result:

THEOREM4. monotone

The discretizatim

matrix A,, E Iw”,”given in (5) is inverse

if

o<-(2,

n+l

o<

S”

and

-574x

-4(n+l)“(l-cos--3

< q

(i = 1,. . . ,n).

< 4(n+1)4sin2stan2-&=47i-4

REMARK. The Green’s function for the boundary-value problem (BVF’) is nonnegative if CY> 0, /I > 0, q E C[O, 11, and - 7r4 < q(s) < 4V4

for all

s E [OJ].

This follows from Theorem 4, since the matrices A, yield a stable discretization for BVP. An elegant way to prove stability is by Gerschgorin’s method [8, 9, 121, where again the inverse monotonicity of A,, is essential.

MATRICES WITH NONNEGATIVE INVERSES

141

REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12

13 14 15 16 17 18 19 20 21 22

It. Belhnann, Introducti to Matrix Analysis, McGraw-Hill, New York, 1960. E. Bohl, On a stability inequality for nonlinear operators, SLAM .J. Numer. Anal. 14:242-25‘3 (1977). E. Bohl and J. Lorenz, Inverse monotonicity and difference schemes of higher order: a summary for two-point boundary value problems, to appear. J. H. Bramble and B. E. Hubbard, New monotone type approximations for elliptic problems, Math Curnp. 18:349-367 (1964). L. Collatz, Aufgaben monotoner Art, Arch. Math. (Basel) 3:365-376 (1952). W. A. Coppel, Discunjugacy, Lecture Notes in Math. 220, Springer, New York, 1971. F. R. Gantmacher and M. T. Krein, Oszillationsmatrizen, Oszillutimkerne und kleine Schtingungen me&mix&r Systeme, Akademieverlag, Berlin, 1960. S. Gerschgorin, Fehlerabschlzung fur das Differenzenverfahren zur Losung partieller Differentialgleichungen, Z. Angew. Math. Mech. 10:373-382 (1930). R. Gorenflo, Uber S. Gerschgorins Methode der Fehlerabschatzung bei Differenzenverfahren, in Lecure Notes in Math. 333, Springer, New York, 1973. P. Henrici, Discrete Variable Methods in Ordinary LIifferential Equutions, Wiley, New York, 1962. S. Karhn, Tot& Positioity, Vol. I, Stanford U. P., Stanford, 1968. J. Lorenz, Die Inversmonotonie von Matrizen und ihre Anwendung beim Stabilitltsnachweis von Differenzenverfahren, Dissertation, Univ. Miinster, 1975. J. Iorenz, Zur Inversmonotonie diskreter Probleme, Numer. Math. 27~227-238 (1977). D. S. Meek, A new class of matrices with positive inverses, Linear Algebra and A&. 15:253-260 (1976). N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polyrwme, Deutscher Verlag der Wissenschaften, 1963. A. Ostrowski, Uber die Determinanten mit iiberwiegender Hauptdiagonale, Comment. Math. Helu. lo:6996 (1937). R. J. Plemmons, M-matrix characterizations. I-Nonsingular M-matrices, Linear Algebra and Appl. 18:175-188 (1977). H. S. Price, Monotone and oscillation matrices applied to finite difference approximations. Math. Camp. 22:489-516 (1968). J. Schroder, Lineare Operatoren mit positiver Inversen, Arch. Rational Mech. Anal. 8:408-434 (1961). J. Schroder, Proving inverse-positivity of linear operators by reduction, Numer. Math. 15:100-108 (1970). W. F. Trench, An algorithm for the inversion of finite Toeplitz matrices, I. Sot. Zd. and Appl. Math. 12:515-523 (1964). R. S. Varga, Matrix lteratiue Analysis, Prentice-Ha& Englewood Cliffs, N.J., 1962. Received6 Februay 1978