Advances in Fuzzy Mathematics. ISSN 0973-533X Volume 12, Number 4 (2017), pp. 855-866 © Research India Publications http://www.ripublication.com
On the Eigenproblems of Nilpotent Lattice Matrices
Geena Joy Department of Mathematics, Union Christian College, Aluva, Kochi, Kerala, India – 683102
K. V. Thomas Department of Mathematics, Bharata Mata College, Thrikkakara, Kochi, Kerala, India – 682021
Abstract
This paper discusses the eigenproblems of nilpotent lattice matrices and describes the eigenspace of a nilpotent lattice matrix. Also, this paper introduces the concept of non-singular lattice matrices. We prove that every invertible lattice matrix is non-singular and every nilpotent lattice matix is singular.
Keywords: Distributive lattice, Nilpotent lattice matrix, eigenspace, Non-singular lattice matrix, Invertible lattice matrix
2010 MSC: 15B99
1. INTRODUCTION The notion of lattice matrices appeared firstly in the work, ‘Lattice matrices’ [2] by G. Give’on in 1964. A matrix is called a lattice matrix if its entries belong to a distributive lattice. All Boolean matrices and fuzzy matrices are lattice matrices. Lattice matrices in various special cases become useful tools in various domains like the theory of switching nets, automata theory and the theory of finite graphs [2]. The concept of nilpotent lattice matrix was introduced firstly by G. Give’on [2]. A m square lattice matrix A is called nilpotent if A = 0n , for some positive integer m , where 0n is the zero matrix. In [2], Give’on proved that an nn lattice matrix A is n nilpotent if and only if A = 0n . Since then, a number of researchers have carried out their research in the area of nilpotent lattice matrices (see [6, 8, 9, 10]). The eigenvector-eigenvalue problems (for short, eigenproblems) of matrices over a 856 Geena Joy and K. V. Thomas complete and completely distributive lattice with the greatest element 1 and the least element 0 (lattice matrices) were discussed firstly by Y. J. Tan [5]. In [7], Y. J. Tan raised an open problem ’How to describe the eigenspace for a given lattice matrix?’. In this paper, we describe the eigenspace of a given nilpotent lattice matrix. Further, we introduce the concept of non-singular lattice matrices. In classical theory of matrices over a field, the theory of non-singular matrices coincides with the theory of invertible matrices. In the case of lattice matrices, every non-singular lattice matrix need not be invertible. Also, we prove that every nilpotent lattice matrix is singular.
2. PRELIMINARIES We recall some basic definitions and results on lattice theory and lattice matrices which will be used in the sequel. For details see [1, 2, 3, 4, 7, 9, 10]. A partially ordered set L ),( is a lattice if for all , Lba , the least upper bound of ba },{ and the greatest lower bound of ba },{ exist in L . For any , Lba , the least upper bound and the greatest lower bound is denoted by ba and ba (or ab ), respectively. An element La is called greatest element of L if ax , Lx . An element Lb is called least element of L if xb , Lx . We use 1 and 0 to denote the greatest element and the least element of L , respectively. A nonvoid subset K of a lattice L is a sublattice of L , if for any , Kba , ba , Kba . A lattice L is called a complete lattice if for any LH , both the least upper bound Hyy }|{ and the greatest lower bound Hyy }|{ of H exist in L . A lattice L ),,,( is a distributive lattice if the operations and are distributive with respect to each other. For any , Lba , the greatest element Lx satisfying the inequality bxa is called the relative pseudocomplement of a in b and is denoted by ba . If for any , Lba , ba exists, then L is said to be a Brouwerian lattice. For La , a 0 is denoted by ac . A lattice L is said to be completely distributive if for any Lx and any
i }|{ LIiy , where I is an index set,
(a). yxyx iIiiIi )(=)( and
(b). yxyx iIiiIi )(=)( holds. It is known [1] that a complete lattice L is Brouwerian if and only if (a) is satisfied in L . Therefore, a complete and completely distributive lattice L is Brouwerian. Unless otherwise specified, we shall assume throughout the paper that L is a complete and completely distributive lattice with the greatest element 1 and the least element 0 , respectively.
Lemma 2.1 [4, 7] Let { i ,1 } Lnia . Then we have c c (a). 1 aini =)( 1 aini c c (b). 1 ini 1 aa ini )( . On the Eigenproblems of Nilpotent Lattice Matrices 857
Lemma 2.2 Let , Lba such that ba cc 0== .Then ab c 0=)( . Proof. If possible, assume that ab c 0>)( . Then there exists y 0> such that ab y 0=)( . That is, there exists y 0> such that ( )(byay 0=) . cc Let y1 = ay and y2 = by . Then yy 21 0>, . Otherwise, ba 0>, . Now by1 b abay y 0=)(=)(= and ay2 a abby y 0=)(=)(= .
Thus there exist yy 21 0>, such that ayby 21 0== , which is a contradiction. Hence ab c 0=)( .
Let n LM )( be the set of all nn matrices over L (Lattice Matrices). We th shall denote by Aij the element of L which stands in the ji ),( entry of n LMA )( .
For (= ij ), (= ij ), ij n LMCCBBAA )()(= , define
ijijij njiBACCBA ),1,2,=,(==
aA CC aAijij ,== nji ),1,2,=,(Lafor n AB == njiBACC (),1,2,=,( Multiplica tion) ij ik kj k 1= T jiij njiACCA (),1,2,=,(== Transposition) 1, if = ji I ij =)( 0n ij nji ),1,2,=,(0=)( 0, if ji 0 1 =,= kk for 0,kkAAAIA is .integeran
It is clear that n LM )( is a semigroup with the identity I and the zero 0n with respect to the multiplication. Also, the transposition in n LM )( has the following properties: (a). AB =)( AB TTT (b). TT =)( AA .
Let n LMA )( . Then A is called nilpotent, if there exists some integer k 1 k such that A = 0n .
n Theorem 2.3 [2] Let n LMA )( . Then A is nilpotent if and only if A = 0n .
Theorem 2.4 [10] Let LMA )( be nilpotent. Then aaa 0= , for n ii 211 iii mi
21 iiii m n},{1,2,},,,,{ .
Let n LMA )( . Then A is said to be invertible if there is a n LMB )( such that BAAB == I . Here B is called the inverse of A and is denoted by A1 . 858 Geena Joy and K. V. Thomas
A set 21 aaaS m},,,{= of elements of L is a decomposition of 1 in L if and only if aimii 1= . The set S is called orthogonal if and only if aa ji 0= , for all ,,1,2,=, jimji . Hence S is called an orthogonal decomposition of 1 in L if and only if it is orthogonal and a decomposition of 1 in L .
Theorem 2.5 [2] Let n LMA )( . Then A is invertible if and only if each row and each column of A is an orthogonal decomposition of 1 in L .
Let n LMBA )(, . If there exists an invertible matrix n LMP )( such that = PB 1AP , then B is said to be similar to A .
For n LMA )( , the permanent of A is defined as | |= aaaA , (2)2(1)1 nn )( Sn where Sn denotes the symmetric group of all permutations of the indices n}{1,2, .
Lemma 2.6 [9] Let n LMA )( be invertible. Then | A|= 1.
Let n LV )( be the set of all column vectors or n -vectors over L . For T T 21 n 21 n n LVyyyyxxxx )(),,,(=,),,,(= and La , define T Addition: 2211 yxyxyxyx nn ),,,(= , scalar multiplication: xa axaxax ),,,(= T and 21 n T n LV )(,0)(0,0,=0 . Then LV )( is a lattice vector space over L . The elements of LV )( are called n n as vectors and the elements of L are called as scalars. Also, 0 is called the zero vector. Denote by e ,1)(1,1,= T . For T T LVyyyyxxxx )(),,,(=,),,,(= , define 21 n 21 n n ii niyxyx ),1,2,=( T 2211 yxyxyxyx nn ),,,(= T 2211 yxyxyxyx nn ),,,(= ccc xxxx ),,,(= Tc 21 n = = xAyxAy , ,1,2,=((L)MAfor ni ). i ij j n 1 nj
ccc Lemma 2.7 [7] For n LVyx )(, , we have =)( yxyx .
T A vector 21 xxxx n ),,,(= is an ortho vector when xx ji 0= , for T ,1,2,=, nji and ji . An ortho vector 21 xxxx n ),,,(= is a stochastic vector when 1 xini 1= . On the Eigenproblems of Nilpotent Lattice Matrices 859
Let W be a non-empty subset of n LV )( . Then W is a sub lattice vector space of LV )( if and only if W is closed under addition and scalar multiplication in LV )( n n . Let S be a subset of W . Then a vector Wv is a linear combination of the vectors in S , if there exists ,,, Svvv and ,,, Laaa such that .= vav . 21 n 21 n 1 iini Assume that there exists WS such that every Wv is a linear combination of the vectors from S . Then S is called a spanning subset of W .
3. EIGENPROBLEMS OF NILPOTENT MATRICES OVER L In this section, we consider the eigenproblems (See [7]) of nilpotent matrices over L , a complete and completely distributive lattice with 1 and 0 .
Definition 3.1 [5] Let LMA )( . An eigenvector of A is a vector x in LV )( n n such that = xxA , for some scalar in L . The element is called the associated eigenvalue.
First, we consider a given eigenvalue of a nilpotent matrix A in n LM )( and discuss the properties of its eigenvectors.
For a given eigenvalue of a matrix A in n LM )( , the set of all eigenvectors of is denoted by EA )( . That is, A n xxALVxE }=|)({=)( .
Theorem 3.2 [7] Let n LMA )( and be a given eigenvalue of A . Then EA )( is a sub lattice vector space of LV )( and is called the -eigenspace of A . Moreover, n EA )( has the least element 0 and the greatest element T n g eAeAex )()(=)( .
Theorem 3.3 Let n LMA )( be nilpotent and be a given eigenvalue of A . Then E )( has the least element 0 and the greatest element eAex )(=)( cT . A g Proof. By Theorem 3.2, EA )( has the least element 0 and the greatest element
T n T g 0n eeAeeAeAex )()(=)()(=)( (By Theorem 2.3) T cT eAeeAe .)(=0)(=
Theorem 3.4 Let n LMA )( be nilpotent and be a given eigenvalue of A . Then
EA )( is spanned by T T T T {(x1 x2 xn },0){(0,0,}),(0,0,,,,0),(0,,,0),0, , T cT where g 21 n eAexxxx )(=),,,(=)( is the greatest element of EA )( . T T T Proof. Let xx 11 2 xx 2 xn xn ),(0,0,=,,,0),(0,=,,0),0,(= . 860 Geena Joy and K. V. Thomas
T Firstly we prove that i xx i EA ,)(,0),,(0,= where x (= A =) cc Ac , for ,1,2,= ni (By Lemma 2.1(a)). i 1 nj ji 1 nj ji To do this, for ,1,2,= ni , cosider T i xAxA i ,0),,(0,=
1211 AAA 1n AAA 2221 2n T = xi ,0),,(0, nn 21 AAA nn T 21 iiii nixAxAxA i ),,,(= A ((= c c ), AA (c c ), AA (, c A ))Tc 1i ji 2i ji ni ji 1 nj 1 nj 1 nj T .0=,0)(0,0,= Also, T T xi xi xi ,0),,(0,=,0),,(0,= (,(0,= c Ac ),,0)T ji 1 nj T .0=,0)(0,0,=
Therefore = xxA ii , for ,1,2,= ni . Hence 21 Exxx An )(,,, . T Now let 21 yyy n ),,,( be any eigenvector of . Then by Theorem 3.3, T T 21 n 21 xxxyyy n ),,,(),,,(
xy ii , .n,1,2,=ifor
Take = ya ii , for ,1,2,= ni . Then i La and = xay iii , for ,1,2,= ni . Therefore T T 21 n 2211 xaxaxayyy nn ),,,(=),,,(
xa T xa ,0),(0,,0),0,(= T xa ),(0,0, T 11 22 n n = 2211 xaxaxa nn . Hence the -eigenspace EA )( is spanned by 21 xxx n }0{},,,{ .
Theorem 3.5 Let LMA )( be nilpotent and be a given eigenvalue of A . Then n c 0= if and only if the only eigenvector of is 0 . Proof. First assume that c 0= . Then by Lemma 2.7, cT cTc eAeAeeAex cT .0=)(0=)()(=)(=)( g Hence by Theorem 3.3, the only eigenvector of is 0 . Conversely assume that the only eigenvector of is 0 . Then by Theorem 3.3, On the Eigenproblems of Nilpotent Lattice Matrices 861 eAex cT A c ni .0,1=)(0=)(=)( g ji 1 nj If possible, assume that c 0 . Then A c 0,1=)( ni A c 0,1)( ni ji ji 1 nj 1 nj c A c 0,1)()( ni ji 1 nj A c 0,1)( ni ji 1 nj c Aji 0, somefor i, n,1,2,=j Let = pi and = pj . Then A c 0 . 1 1pp Now ( A c (0=) AA c 0) jp1 jp 11 pp 1 nj 1 nj A c AA c 0)()( 1pp jp 11 pp 1 nj AA c 0)( jp 11 pp 1 nj AA c 0, somefor ,1,2,=j n, )p(say jp 11 pp 2 Hence AA c 0 . 112 pppp Proceeding as above, we get AAA c 0, where1 p, .np,,p,p pn pn 1121 pppp n21 But, by Theorem 2.4, AAA c = 0, where1 p, .np,,p,p pn pn 1121 pppp n21 Therefore, we get a contradiction. Hence c 0= .
Next, we consider a given eigenvector x of a nilpotent matrix A in n LM )( and discuss the properties of its eigenvalues. For , Lba , the interval bxaLxba }|{=],[ .
Theorem 3.6 [5] Let n LMA )( and x be an eigenvector of A . Then the set of all eigenvalues of x form a sublattice of L consisting of the interval ul ],[ , where = Tl xAe and = Tu xe l .
A modification for the value of l in Theorem 3.6 can be given as = Tl xAx , since Tl T xxxAx ,== somefor Lin T TT T xAexexexx .==== 862 Geena Joy and K. V. Thomas
Theorem 3.7 Let n LMA )( be nilpotent and x be an eigenvector of A . If x is an ortho vector of A with xe cT 0=)( , then x has the unique eigenvalue 0 . Proof. We have by Theorem 2.4, Tl == xxAxAx = xxA 0= and ,1 nji ij ji ,,1 jinji ij ji = Tu Tl xexexe cT 0=)(=0= . Hence by Theorem 3.6, x has the unique eigenvalue 0 .
Corollary 3.8 Let LMA )( be nilpotent and x be an eigenvector of A . If x is n a stochastic vector of A , then x has the unique eigenvalue 0 .
Theorem 3.9 [7] If an eigenvector x of n LMA )( has the unique eigenvalue , then 0 nT eAe .
Theorem 3.10 Let n LMA )( be nilpotent. If an eigenvector x of A has the unique eigenvalue , then 0= . n Proof. Since A is nilpotent, by Theorem 2.3, we have A = 0n . Therefore by T Theorem 3.9, 0 0n ee 0= . Hence 0= .
Theorem 3.11 Let LMA )( be nilpotent and x be an eigenvector of A with the n associated eigenvalue 0 , then x is an eigenvector of A with the associated eigenvalue 0 . Proof. Let x be an eigenvector of A with the associated eigenvalue 0 , then = xxA . By Theorem 2.3, we have nn 1 n1 ======0 xxxAxAxAxAAxA . Therefore, 0=0== xxxA . Hence the proof is complete.
4. NON-SINGULAR LATTICE MATRICES In this section, we introduce the concept of non-singular matrices over a complete and completely distributive lattice with 1 and 0 .
Definition 4.1 Let n LMA )( . If the sub lattice vector space A n xALVxE }0=|)({=(0) of n LV )( contains a non-zero vector, then A is said to be singular. Otherwise, A is non-singular.
Theorem 4.2 Let LMA )( . Then A is non-singular if and only if eA cT 0=)( . n Proof. We have A is non-singular E }0{=(0) . By Theorem 3.2, A T n T cT g =(0) (0 ) (0 0=) eAeAeAex eA 0=)(0=0 . On the Eigenproblems of Nilpotent Lattice Matrices 863
Theorem 4.3 Let n LMA )( . If A is invertible, then A is non-singular. Proof. Assume that A is invertible. Then by Theorem 2.5, T = eeA . Therefore, eeA ccT 0==)( . Hence by Theorem 4.2, A is non-singular.
However, the converse need not be true. The following example shows that if A is non-singular, then A need not be invertible.
Example 4.4 Consider the lattice ihgfL ,1},,,{0,= , whose diagrammatical representation is as follows:
It is easy to see that L is a distributive lattice.
fh cT c Let A = 2 LM )( . Then hheA 0=),(=)( . Hence by Theorem 4.2, hf
A is non-singular. But by Theorem 2.5, we can see that A is not invertible.
Theorem 4.5 Every nilpotent lattice matrix is singular. c Proof. Let n LMA )( be nilpotent. Since 1=0 , by Theorem 3.5, EA (0) contains a non-zero vector. Hence A is singular.
Theorem 4.6 Let n LMBA )(, such that A is non-singular and B is invertible. Then we have (a). A is non-singular if and only if AB is non-singular. (b). A is non-singular if and only if BA is non-singular. Proof. (a) First assume that A is non-singular. Then by Theorem 4.2, eA cT 0=)( . Let C = AB . Then ((AB eCe cTcT =)(=)) (( C c C c C Tc .))(,,)(,) 1 knk 11 11 knk 22 2 1 kn kn nn 864 Geena Joy and K. V. Thomas
Consider C c C c C )()()( c 1 knk 11 1 1 knk 22 2 1 kn kn nn C c C c C c )()()(= Lemma(By 2.1(a)) 1 knk 11 11 knk 22 2 1 kn kn nn = ( c CC c Cc ) k11 k2 2 knn 1 1, 2 ,, kkk n n ( CC C )c Lemma(By 2.1(b)) 11 kk 2 2 knn 1 1, 2 ,, kkk nn ( CC C )c Lemma(By 2.1(a)) 11 kk 2 2 knn 1 1, 2 ,, kkk nn Now CCC 11 kk 2 2 knn (= BA () BA () BA ) 1 iikni 1111 11 iikni 2222 2 1 in kn nin inn (= () BBBAAA ) 2211 kikik nin 11 2 2 iii nn 1 1, 2 ,, iii nn
(= () BBBAAA ) (By Theorem 2.5) 1 (1) kk 2 (2) kn n)( (1)1 (2)2 )( nn Sn
Therefore, C c C c C )()()( c 1 knk 11 1 1 knk 22 2 1 kn kn nn ( CC C )c 11 kk 2 2 knn 1 1, 2 ,, kkk n n ((( () BBBAAA )))c 1 (1) kk 2 (2) kn n)( (2)2(1)1 )( nn 1 1, 2 ,, kkk n Sn n (( BBB () AAA ))c (2)2(1)1 )( nn 11 kk 2 2 knn Sn 1 1, 2 ,, kkk n n ( AAA )c Lemma(By 2.6) 11 kk 2 2 knn 1 1, 2 ,, kkk nn (( A ))c 1 nj ji 1 ni 0 (By Theorem 4.2 and Lemma .2.2)
Thus C c 0=)( , for ,1,2,= ni . Therfore, ((AB e cT 0=)) . Hence AB is 1 ki kn ii non-singular. Conversely assume that AB is non-singular. Then by first part AB 1 AB BB 1 =)(=)( A is non-singular. On the Eigenproblems of Nilpotent Lattice Matrices 865
(b) First assume that A is non-singular. Then by Theorem 4.2, eA cT 0=)( . We have by Theorem 2.5, ((BA cT =)) (( ((=)) )) eAeBAeBAe cTcTTcTT 0=)(= Hence BA is non-singular. Conversely assume that BA is non-singular. Then by first part B1 BA 1 =)(=)( AABB is non-singular.
Theorem 4.7 Let n LMA )( . Then A is non-singular if and only if every lattice matrix similar to A is non-singular.
Proof. Let n LMB )( such that B is similar to A . Then there exists an invertible 1 matrix n LMP )( such that = PB AP . By Theorem 4.6, A is non-singular if and only if B is non-singular. Hence the proof is complete.
5. CONCLUSION In this paper, we partially solved the open problem ’How to describe the eigenspace of a given lattice matrix?’, for a given nilpotent matrix over a complete and completely distributive lattice with 1 and 0 . Also, we intoduced the concept of non-singular lattice matrices. We hope this research would be able to throw light on at least a few new concepts in lattice matrices including Boolean and fuzzy matrices and its applications which will further enrich the subject in science and technology.
ACKNOWLEDGEMENT The first author is thankful to the University Grants Commission, New Delhi, India for the award of Teacher Fellowship under the XII Plan period.
REFERENCES [1] G. Birkhoff, Lattice Theory, 3 rd ed., Amer. Math. Soc. Providence, RI, 1967. [2] Y. Giveón, Lattice matrices, Inform. Control 7 (1964) 477-484. [3] G. Grätzer, General Lattice Theory, 2 nd ed., Birkhäuser, Basel, 1998. [4] E. E. Marenich, Determinant theory for lattice matrices, Journal of Mathematical Sciences 193 (2013) 537-547. [5] Y.J. Tan, Eigen values and eigenvectors for matrices over distributive lattices, Linear Algebra Appl. 283 (1998) 257-272. [6] Y.J. Tan, On the powers of matrices over a distributive lattice, Linear Algebra Appl. 336 (2001) 1-14. [7] Y. J. Tan, On the eigenproblem of matrices over distributive lattices, Linear Algebra Appl. 374 (2003) 87-106. [8] Y. J. Tan, On nilpotent matrices over distributive lattices, Fuzzy Sets and Systems 151 (2005) 421-433. 866 Geena Joy and K. V. Thomas
[9] K. V. Thomas and G. Joy, A Study on Characteristic Roots of Lattice Matrices, Journal of Mathematics, vol. 2016, Article ID 3964351, 8 pages, 2016. doi:10.1155/2016/3964351.
[10] K. L. Zhang, On the nilpotent matrices over D01 - lattice, Fuzzy Sets and Systems 117 (2001) 403-406.