Mathematical Programming 61 (1993) 351-356 351 North-Holland

Ky Fan's N-matrices and linear complementarity problems

Jianming Miao RUTCOR, Rutgers University, New Brunswick, N J, USA

Received 16 March 1992 Revised manuscriplL received 17 July 1992

We consider the linear complementarity problem (LCP), w = Az + q, w >~ 0, z >~ 0, wTz = 0, when all the off- diagonal entries of A are nonpositive (the class of Z-matrices), all the proper principal minors of A are positive and the determinant of A is negative (the class of almost P-matrices). We shall call this the class of F-matrices. We show that if A is a Z-matrix, then A is an F-matrix if and only if LCP(q, A) has exactly two solutions for any q~>0, q#0, and has at most two solutions for any other q.

Key words: Classes of matrices, linear complementarity problem.

1. Introduction

Let A be an n x n ma t r ix wi th real entries. A is cal led a P-matrix (N-matrix) i f all its

p r inc ipa l minor s are posi t ive (negat ive) . A n N - m a t r i x A is said to be o f the first category i f it has at least one posi t ive element, o therwise it is said to be o f the second category (see, e.g. [9]). A is cal led an almost P-matrix i f all its p r o p e r pr inc ipa l minor s

are posi t ive and its de t e rminan t is negative. A is said to be a Z-matrix if all its off-

d i agona l entr ies are nonposi t ive .

R e m a r k 1. A is an a lmos t P -ma t r ix if and only i f A -1 is an N-mat r ix .

T h r o u g h o u t this p a p e r all inequal i t ies between matr ices or vectors o f the same

size, such as A >i B, are componentwise .

F o r a given ma t r ix A e N.x n, and a vec tor q~ Nn, the linear complementarity prob-

lem, deno t ed by LCP(q , A) , is to f ind vectors w, z~ Nn such tha t

w = A z + q , (1.1)

w~>0, z~>0, (1.2)

wTz=O. (1.3)

A pa i r (w, z) o f vectors sat isfying (1 .1)- (1 .3) is cal led a so lu t ion to LCP(q , A) .

Correspondence to : Prof. Jianming Miao, RuTcOR Rutgers Center for Operations Research, Rutgers University, P.O. Box 5062, New Brunswick, NJ 08903-5062, USA.

Research supported by AFOSR-89-0512.

352 J. Miao / Ky Fan's N-matrices and LCP

In this paper, we consider the characterization of a special class of matrices A in terms of the number of solutions to LCP(q, A) for any vector q. A well-known result in this regard is that of Samelson, Thrall and Wesler [13] and Murty [8], which states that A is a P-matrix if and only if LCP(q, A) has a unique solution for every q. Kojima and Saigal [6] studied the class of N-matrices. They proved the following theorem.

Theorem 1. Let A be an N-matrix. I f A < O, then LCP(q, A) has exactly two solutions for any q > 0 and no solution for any q~O. I f A¢O, then LCP(q,A) has a unique solution for any q~O and exactly three solutions for any q> 0 which is nondegenerate with respect to A. []

Remark 2. If A < 0, then LCP(q, A) has a unique solution (q, 0) for any q ~> 0, q ;¢ 0.

The above result was further improved by Gowda [4], see also Section 6.6 of [2]. Recently the converse of the above theorem, i.e., a characterization of N-matrices using the number of solutions to LCP(q, A) was obtained by Parthasarathy and Ravindran [11] (for N-matrices of the second category) and Mohan and Sridhar [7] (for N-matrices of the first category).

Our aim is to add another class of matrices, Ky Fan's N-matrices [3], i.e., the ones that have the form A = a I - B with B~>0 and 2.< a <p(B) , where I denotes the identity matrix, p(B) is the spectral radius of B, and 3~ is the maximum of the spectral radii of all principal submatrices of B of order n - 1. To avoid confusion with our previous definition of N-matrices, we shall call this the class of F-matrices. An equiva- lent definition for F-matrices is as follows [3].

Definition 1.1. A is called an F-matrix if it is a Z-matrix as well as an a lmost P-

matrix.

Our main result is the following: Let A be a Z-matrix. Then A is an F-matrix if and only if LCP(q, A) has exactly two solutions for any q ~> 0, q # 0, and has at most two solutions for any other q.

2. Main results

F-matrices are closely related to M-matrices. A is called an M-matrix if A - - - a I - B with a > p(B), where B is a nonnegative matrix. Equivalently, A is an M-matrix if and only if it is a Z-matrix as well as a P-matrix. For more equivalent conditions about M-matrices, see for example Berman and Plemmons [1]. It follows from the definition of F-matrices that A is an F-matrix if and only if all its principal

J. Miao / Ky Fan's N-matrices and LCP 353

submatrices o f order n - 1 are M-matrices and its determinant is negative. For more results about F-matrices, see Ky Fan [3] and Johnson [5].

To state the next theorem, we need more definitions. We denote by Q the class of matrices such that LCP(q, A) has a solution for every q. We call S a signature matrix if S is a diagonal matr ix with diagonal entries + 1 or - 1 .

Remark 3. I f A is nonsingular, then A e Q if and only if A -~ e Q.

Theorem 2. Let A be a nonsingular Z-matrix. Then the following statements are equivalent:

(i) A is an F-matrix. (ii) A -1 < 0.

(iii) All the diagonal entries of A are positive. For every signature matrix S ¢ 1 or - I , SAS~Q, and Aq~Q.

(iv) At least one diagonal entry of A is nonnegative. For every signature matrix S # I or - I , SAS~Q, and Aq~Q.

Proof. (i) ~ (ii) : see Theorem 3 o f [3]. (ii) ~ (i): see Corol lary 2.8 o f [5]. (i), (ii) ~ (iii) : By Remark 1 and (ii), A -1 is an N-matr ix of the second category,

and SA-IS is an N-mat r ix o f the first category. It follows f rom Theorem 1 that A-l q~Q and SA-1SeQ. Which implies A(~Q and SASeQ.

(iii) ~ (iv): This is obvious. (iv) ~ (i) : Wi thout loss of generality, let a,n >~ 0, and

c) A =

d T ann

Consider any

Let

Then

(qn 1) q= , q n - l E ~n-1 , qnn~, with qn-l<O, q~n>O.

\ qnn

1 1°)

B

w~>0, z~>0, (2.2)

wTz = 0, (2.3)

354

has a solution, where

w=(Wn-1 I \ Wnn /

Therefore

J. Miao / Ky Fan "s N-matrices and LCP

and z = ( Z " - l ) . \ Znn

w,n = -drzn_ l + annznn + qnn) qn, > O, (2.4)

which implies

zn,=0, and Bzn_l=Wn_l-qn_l>/-qn_l>O. (2.5)

Consequently, B is an M-matrix (see for example Theorem 6.2.3 of [1]), and more- over a~ > 0 for i= 2 . . . . , n. Now any a~, i= 2 . . . . , n, can play the role of ann. We conclude that all (n - 1) x (n - 1) principal submatrices of A are M-matrices. A cannot be an M-matrix since A ¢ Q, it must be an F-matrix. []

Theorem 3. Let A be an F-matrix. Then LCP(q, A) has exactly two solutions for any q~>0, q ¢ 0 , no solution for any q <~ O, q #O, and has at most two solutions for any other q.

Proof. LCP(q, A) is equivalent to LCP( -A- Iq , A 1) in the sense that every solution (w, z) to LCP(q, A) corresponds to a solution (z, w) to LCP(-A-1q, A-l), and vice versa. By Remark 1 and Theorem 2, A -1 is an N-matrix and A -1 <0. The result follows from Theorem 1 and Remark 2. []

Theorem 4. Let A be a Z-matrix. I fLCP(q, A) has exactly two solutions for any q >~0, q#O, and has a finite number of solutions for any other q, then A is an F-matrix.

Proof. Since LCP(q, A) has a finite number of solutions for any q, all the principal minors of A are nonzero, see [8]. We shall prove A -1 ~<0. Suppose on the contrary, there exists an integer i such that A-lei ~ O, where e; is the ith column of the identity matrix. Let q = e~, then (e~, 0) is a trivial solution to LCP(e;, A). By our assumptions, there is another solution (w, z) to LCP(ei, A) such that w ¢ 0, z ¢ 0. Let J = {j: zj > 0}, and J = {1, 2 . . . . . n}\J. Then J ¢ 0 , J # 0 , and wj=0, wj¢0 . Without loss of gen- erality, let A be partitioned as

Ajj Asj I (2.6) A = k A j j Asj}"

There are only two possible situations: i e J or ie.7. It will be shown separately in what follows that either situation entails a contradiction, thereby proving A -1 <~0.

J. Miao / Ky Fan's N-matrices and LCP 355

Case 1. i~ J. 'Then we have

:: 0 Therefore we= A: j z j >>, O, w j # O. However Ajj ~< 0 and zj > 0 imply w j = Ajjzs <~ 0, a

contradiction. Case 2. i~ Z Then

Therefore Aj j z : = O, which implies Ajj is singular, a contradiction.

Next we shall show A is irreducible. Suppose on the contrary that A is reducible, i.e., there is a permutat ion matrix P such that

where B and D are square matrices. Without loss of generality, let A be partit ioned a s

z q l It follows from A-1 ~< 0 that B - 1 ~< 0. Now consider any q (q2) > 0 partitioned con- formally with A. Then LCP(q, A) has at least three solutions.

(1) w--q, z = 0 .

(2) w~-O, z = - A - l q .

(o) olq) ( 3 ) w = , z = ( . q2

This is a contradiction to our hypothesis. In summary, we have shown that all the principal minors of A are nonzero, A - ~ 0 and A is irreducible. It follows from Theorem 2.7 of [5] that A is an F-matrix. []

Combining Theorem 3 and Theorem 4 we have our main result.

Theorem 5. Let A be a Z-matrix. Then A is an F-matrix i f and only t fLCP(q, A) has exactly two sohttions for any q>~O, q # 0 , and has at most two solutions for any other q. []

356 J. Miao / K y Fan's N-matrices and LCP

Acknowledgement

The author is grateful to Professor Adi Ben-Israel for his valuable suggestions. After this work was completed, I have been informed by one referee that a slightly

different version of Theorem 5 was also obtained by Xu [ 14 ].

References

[1] A. Berman and R.J. Plemons, Nonnegative Matrices in the Mathematical Sciences (Academic Press, New York, 1979).

[2] R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem (Academic Press, Boston, MA, 1992).

[3] K. Fan, "Some matrix inequalities," Abhandlungen aus dem Mathematischen Seminar der Universitiit Hamburg 29 (1966), 185-196.

[4] M.S. Gowda, "Applications of degree theory to linear complementarity problems," Research Report 91-14, Department of Mathematics, University of Maryland Baltimore County. (Catonsville, MD, 1991).

[5] G.A. Johnson, "A generalization of N-matrices," Linear Algebra and its Applications 48 (1982) 201-217.

[6] M. Kojima and R. Saigal, "On the number of solutions to a class of linear complementarity problems," Mathematical Programming 17 ( 1979 ) 136-139.

[7] S.R. Mohan and R. Sridhar, "On characterizing N-matrices using linear complementarity," Linear Algebra and its Applications 160 (1992) 231 245.

[8] K.G. Murty, "On the number of solutions to the complementarity problem and spanning properties of complementary cones," Linear Algebra and its Applications 5 (1972) 65-108.

[9] H. Nikaido, Convex Structures and Economic Theory (Academic Press, New York, 1968). [10] C. Olech, T. Parthasarathy and G. Ravindran, "Almost N-matrices and linear complementarity,"

Linear Algebra and its Applications 145 (1991) 107-125. [11] T. Parthasarathy and G. Ravindran, "N-matrices," Linear Algebra and its Applications 139 (1990)

89-102. [ 12] R. Saigal, "On the class of complementary cones and Lemke's algorithm," SIAM Journal on Applied

Mathematics 23 (1972) 46 60. [13] H. Samelson, R.M. Thrall and O. Wesler, "A partition theorem for Euclidean n-space," Proceedings

o f the American Mathematical Society 9 (1958) 805-807. [14] S. Xu, "Sign-reversal property for F-matrices and N-matrices," unpublished manuscript.