Iteratioal Joural of Algebra, Vol. 2, 2008, o. 4, 197-204 O Nosigularity of Saddle Poit Matrices with Vectors of Oes Tadeusz Ostrowski Istitute of Maagemet The State Vocatioal Uiversity -400 Gorzów, Polad Tadostr@Iteria.pl Abstract This paper describes ad aalyzes osigularity coditios of saddle poit matrices with two vector blocks of oes. These kid of matrices occur i stadard quadratic program (whe the left-up block is a symmetric matrix) ad i game theory (whe the leftup block is ay square matrix) or i special liear systems called equilibrium systems. Necessary ad sufficiet coditios for the osigularity are ivestigated. We review some kow coditios i the literature as well. Mathematics Subject Classificatios: 15A03, 15A15 Keywords: Saddle Poit Matrix, Nosigularity Coditio 1. Itroductio The paper deals with saddle poit matrices with two vector of oes, i.e. the block matrices i which the top-left submatrix is a real square matrix, the top-right block is the colum vector e (the vector with all compoets 1), the bottom-left block is e trasposed, ad the bottom-right block has the sigle etry 0. If the top-left submatrix is symmetric, the the saddle poit matrix may be iterpreted as the bordered Hessia of a stadard quadratic program over the stadard simplex ad it is usually called the Karush-Kuh-Tucker matrix of the program which is kow to have a very large spectrum of applicatios (for a review, see Bomze [2]). It is well kow that it is ot ecessary for the left-up submatrix to be defiite or eve osigular for the saddle poit matrix to be osigular.
198 T. Ostrowski 2. Notatio Deote by e R the colum vector with all oes, ad let E R, stads for the matrix with all etries equal to oe. Hece E = ee T. For a give matrix A R, deote by (A, e) the matrix, ofte called the saddle poit matrix, havig the structure (A, e) = A T e 0. (1) Let Z R, 1 deotes a matrix give as follows Z = 2 2 2 2 O 3......... ( 1) ( 1) ( 1) ( 1) ( 1) ( 1)... ( 1). (2) Etries of colums of Z sum up to zero ad its orthoormal colums are a basis of the ull space of e T. It ca be writte shortly e T Z = 0, Z T Z = I. A B Block matrices are ofte deoted as the ordered quadruple (A, B, C, D). C D I this paper, sice always we cosider the case B = C T = e, ad D = 0, the the shorteed otatio (A, e) is used. By (A k, e), (A k, e); k = 1,, is deoted the matrix obtaied from (A, e) by replacig i A the k-th row (k-th colum, respectively) with the vector e T (e, respectively). The stadard otatio M/A is used for the Schur complemet of the osigular matrix A i the partitioed matrix M. Recall that two square matrices A ad B are called cogruet, which is writte as A B, if there exists a osigular matrix P such that B = P T AP. 3. Properties of (A, e) I this sectio we preset a few basic properties of the saddle poit matrices with the structure (1). More the properties ca be foud i Ostrowski [5]. Let start from the followig: Theorem 1. If A, E = ee T R, the for ay real umbers α, β the followig holds:
Nosigularity of saddle poit matrices 199 det(αa + βe, e) = α 1 det(a, e) (3) det(αα + βe) = α 1 [α deta β det(a, e)] (4) Proof. Obviously if α = 0, the (3) holds trivially. Assumig that α 0, subtractig the last colum multiplied by β from each of the other colums i the matrix (αa+βe, e) ad the usig multiliearity of the determiat, gives αa + βe e αa e det(αa+βe, e) = det T = det e 0 = T 0 = a 1 A e α det T = a 1 det A e 0 = α 1 det(a, e). T 0 To prove (4) we show first that for ay B R, the particular equality is true, amely det(b + βe) = detb β det(b, e), (5) which trivially holds for β = 0. The derivative of det(b + βe) with respect to β gives d[det( B + βe)] = det (B + βe) i = det B i = det(b, e). dβ i = 1 i = 1 Sice the, we have det(b + βe) = det (B, e)dβ = βdet(b, e) + C. Substitute β = 0, we get C = detb, which gives (5). The geeral case of (4) is obtaied from (5) by replacig B with αa: det(αb + βe) = det(αa) β det(αa, e) = α deta βα 1 det(a, e) = = α 1 [α deta β det(a, e)]. Theorem 2. Let A = (a ij ) R,. If all etries of every row A sum up to the same costat c R, the det(a k, e) = 1 det(a, e). () If all etries of every colum of A sum up to the same costat c R, the det(a k, e) = 1 det(a, e). (7) I each cases det(a, e) = deta, c 0 c det( A βe), c = 0, β 0 β (8)
200 T. Ostrowski I particular if c = 0 the det(a βe) = βdet(a, e) for all β. Proof. We start from the equatio () for c = 0. By assumptio, i the cosiderable case the followig equality holds a kj = i k a. ij Deote by M ij (i, j = 1,, ) miors of the matrix A. It ca be cocluded that the miors with respect to the k-th row are equal ad they differ oly i sig, i.e. M kj + M k,j+1 = 0 The equality above meas that all cofactors of A are equal. Therefore det(a k, e) = det(a k+1, e), k = 1,, 1. Hece it implies () for c = 0. The proof of (7) for c = 0 rus similarly. The equality (8) for c = 0 follows from Laplace evaluatio ad (4), if we put α = 1, β 0. Assume ow that the etries of all rows of the give matrix A sum up to c 0 ad deote by à = (ã ij ) a matrix with etries defied as follows ã ij = a ij c for all i, j = 1,,. Therefore the matrix à has the form give as follows à = A c E ad obviously etries of à sum up zero. Sice the det à = 0. O the other had, by (4), we have det à = det(a c E) = deta + c det(a, e), ad this meas that the equality (8) holds for ay c R. We ca show ow () for ay c R. Sice () is true for c = 0, we have a kj j = 1 ~ = 0 => det(ã k, e) = 1 det(ã, e). (9) e It is obvious that by property of the determiat, if we add last colum multiplied by to each colum except the k-th oe, we get c det(ã k, e) = det(a k, e). Therefore the right side of (9) ca be ow rewritte as follows det(a k, e) = 1 det(a c E, e). Settig α = 1, β = c i (3), we get (). Aalogically we prove (7). Theorem 3. If A is osigular, the the followig holds
Nosigularity of saddle poit matrices 201 raka, if e T A 1 e = 0, rak(a, e) = (10) 1 + raka, if e T A 1 e 0. Proof. By Schur determiat formula we obtai det(a, e) = e T A 1 e det A. O the other had, Guttma rak additivity formula, Zhag [7], states that for a matrix (A, e), if A is osigular, the rak(a, e) = raka + rak(e T A 1 e), which eds the proof. Corollaries. If (A, e) is osigular ad rak(a, e) = + 1, the the raka is either or 1. The coverse, of course, is ot true. However, if the matrix A is both symmetric ad positive defiite, the det(a, e) 0 (see also [1]). Moreover, if A is positive defiite, the (A, e) has oe egative eigevalue ad positive oes. The reaso is that if A is symmetric, the (A, e), beig a Hessia of appropriate Lagragea, is also symmetric but ievitably idefiite. If A is symmetric ad osigular, the (A, e) A 0 s, where s = (A, e)/a = e T A 1 e, ad both cogruet matrices have equal determiat (see also []). Really, for P = I 1 A e 1 we have (A, e) = P T A 0 s P. The cogruece above implies that if A is positive defiite, the the uique egative eigevalue of (A, e) is equal to s. 4. Characterizatio of Nosigularity Various differet coditios for osigularity of matrices with the structure (1) have bee formulated at several places i the literature. Gasterer, Scheid ad Ueberhuber, [3] state that the followig properties are equivalet for give (A, e) R +1, +1 : (i) rak(a, e) = + 1,
202 T. Ostrowski (ii) det[az e] 0, (iii) det(z T AZ) 0, (iv) A T A + E 1 A T EA is positive defiite. Recall also ecessary coditio for osigularity of (A, e). It is give as follows (see for istace, Bezi, Golub ad Liese [1]): A det(a, e) 0 => rak T =. Theorem 4. Let A R,. For the followig statemets (i) rak(a, e) = + 1, (v) det(a + βe) 0 for some β R, (vi) e T x = 0 = > x T Ax 0 (x 0), the followig equivaleces hold: (1) <=> (2) for ay matrix A, (1) <=> (3) for symmetric positive semidefiite matrices, Proof. (i) => (v): From equality (4), settig α = 1, we have det(a + βe) = deta βdet(a, e). Therefore, sice det(a, e) 0, holds det(a + βe) 0 for β R {0}, if deta = 0, for β det A, if deta 0. det( A, e) (v) => (i): Let β be a ozero real umber such that det(a + βe) deta + det(βe) = deta. Applyig (4) for α = 1, we get βdet(a, e) = deta det(a + βe) 0, which eds the proof. (i) => (vi) Proof by cotradictio. Suppose that for some ozero vector x R holds e T x = 0 ad x T Ax = 0. The suppositio that A is positive semidefiite implies that Ax = 0 (see for i-
Nosigularity of saddle poit matrices 203 x stace Hor ad Johso [4]). Sice the, settig v = 0, we have A e x (A, e)v = T 0 = 0. Therefore rak(a, e) < +1 (cotradictio). (vi) => (i). We have to show that (A, e)v = 0 => v = 0. x Let set v = ; x R, α R. The equality (A, e)v = 0 meas that the followig α equivalece holds A e x 0 T 0 = α <=> (e T x = 0 i Ax + αe = 0). From above multiplyig the equality Ax + αe = 0 by vector x T, we get x T Ax + αx T e = x T Ax + α(e T x) T = x T Ax + α0 = x T Ax = 0. It implies by suppositio that x = 0. Therefore we obtai that αe = 0, ad cosequetly α = 0, ad fially v = 0. Note that (A, e) with A symmetric semidefiite, without coditio (vi), is ot 1 1 ecessarily osigular. For example, if A =, the det(a, e) = 0. 1 1 Note also that i geeral A eed ot be osigular for (A, e) to be osigular, but if A is osigular (Theorem 3), the rak(a, e) = raka, if e T A 1 e = 0, 1 + raka, if e T A 1 e 0. Refereces [1] M. Bezi, G. H. Golub ad J. Liese. (2005). Numerical solutio of saddle poit problems, Acta Numerica, Cambridge Uiversity Press, pp.1 137. [2] I. M. Bomze. (1998). O stadard quadratic optimizatio problems, Joural of Global Optimizatio 13, pp.39 387. [3] W. Gasterer, J. Scheid ad C. Ueberhuber. (2003). Mathematical Properties of Equilibrium Systems, Techical Report AURORA TR2003-13.
204 T. Ostrowski [4] R, A. Hor ad C. R. Johso. (1985). Matrix Aalysis. Cambridge Uiversity Press, Cambridge. [5] T. Ostrowski. (2007). O Some Properties of Saddle Poit Matrices with Vector Blocks, Iteratioal Joural of Algebra, Vol.1, o.3, pp. 129 138. [] Y. Tia ad Y. Takae. (2005). Schur complemets ad Baachiewicz-Schur forms. Electroic Joural of Liear Algebra, 13, pp. 405-418. [7] F. Zhag. (2005), Editor. The Schur Complemet Ad Its Applicatios. Numerical Methods ad Algorithms, Vol. 4, Spriger-Verlag, New York Ic. Received: September 1, 2007