Abstract
This paper deals with the characteristic roots of different types of lattice matrices and proves that a matrix and its transpose have the same characteristic roots. Also this paper introduces the concept of similar lattice matrices and proves that similar lattice matrices have the same characteristic roots.
1. Introduction
The notion of lattice matrices appeared firstly in the work “Lattice Matrices” [1] by Giveón 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 [1].
In classical theory of matrices over a field, the characteristic equation of the matrix concerned can be defined determinantally. In the case of lattice matrices, this is not possible. In 1994, Zhang [2] proved the generalization of the Cayley-Hamilton theorem for lattice matrices. Corresponding lattice equation is taken as the characteristic equation of lattice matrix. For matrices over a field, the eigenvector-eigenvalue problems [3] are closely associated with the characteristic equation of the matrix concerned. However, this relationship is vague for lattice matrices.
The roots of characteristic equations of matrices over lattices were studied firstly in the work of Tan [4]. In [4] Tan proved the existence of the least element and the greatest element for the set of all roots of the characteristic equation of a matrix over complete and completely distributive lattices with 0 and 1.
In the present work, we continue to discuss roots of characteristic equations of matrices (based on [4] by Tan) over a class of complete and completely distributive lattices with 0 and 1. Such lattices are abundant; for example, every finite distributive lattice, a complete chain, and, especially, the real interval are all such kind of lattices. In Section 3, we study characteristic roots of different types of lattice matrices. We also introduce the concept of similar lattice matrices and prove that similar lattice matrices have the same characteristic roots.
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, 3–7].
A partially ordered set is a lattice if for all , the least upper bound of and the greatest lower bound of exist in . For any , the least upper bound and the greatest lower bound are denoted by and (or ), respectively.
A nonvoid subset of a lattice is a sublattice of , if for any , , . A lattice is called a complete lattice if for any , both the least upper bound and the greatest lower bound of exist in . A lattice is a distributive lattice if the operations and are distributive with respect to each other.
An element is called greatest element of if , . An element is called least element of if , . We use 1 and 0 to denote the greatest element and the least element of , respectively.
For any , the least element satisfying the inequality is called the relative lower pseudocomplement of in and is denoted by . If for any , exists, then is said to be a dually Brouwerian lattice.
A lattice is said to be completely distributive if for any and any , where is an index set,(a),(b) holds. It is known [5] that a complete lattice is dually Brouwerian if and only if (b) is satisfied in . Therefore, a complete and completely distributive lattice is dually Brouwerian.
In this paper, denotes a complete and completely distributive lattice with the greatest element 1 and the least element 0, respectively.
Lemma 1 (see [7]). Let be a complete and completely distributive lattice with 1 and 0. Then for any , one has (a),(b),(c); in particular, .
Let be the set of all matrices over (lattice matrices). We shall denote by or the element of which stands in the entry of .
For , define , , , , , , ,, for , is an integer. .
Lemma 2 (see [1]). For any , one has the following.(a)The multiplication in has the following properties:(i),(ii),(iii).(b)The transposition in has the following properties:(i),(ii).
Let . If , then is called idempotent; if there exists some integer such that , then is called nilpotent; if , , , then is called upper triangular; if , , , then is called lower triangular; if is both upper triangular and lower triangular, then is called diagonal: that is, ; if is diagonal and , , then is called a scalar matrix.
Theorem 3 (see [1]). Let . Then is nilpotent if and only if .
Let . Then is said to be invertible if there is a such that . Here is called the inverse of and is denoted by . Also, is called orthogonal if and only if .
Theorem 4 (see [1]). Let . Then is invertible if and only if is orthogonal. Hence, .
A set of elements of is a decomposition of 1 in if and only if . The set is called orthogonal if and only if , for all , . Hence, is called an orthogonal decomposition of 1 in if and only if it is orthogonal and a decomposition of 1 in .
Theorem 5 (see [1]). Let . Then is invertible if and only if each row and each column of is an orthogonal decomposition of 1 in .
For , the permanent of is defined as where denotes the symmetric group of all permutations of the indices .
Example 6. For ,
Theorem 7 (see [2]). Let . Then
Theorem 8 (see [8]). Let such that and are invertible. Then
Let denote the set of all column vectors (-vectors) over . It can be considered as an lattice matrix over . Denote .
Let . An eigenvector of is a vector in such that , for some scalar in . The element is called the associated eigenvalue.
3. Characteristic Roots of Matrices over
In this section, we discuss characteristic roots of different types of matrices over a complete and completely distributive lattice with 0 and 1.
Theorem 9 (see [2]). Let . Then ∨, where is the symmetric group of all permutations of the set , , and , when is even; and , when is odd.
In 1994, Zhang [2] proved Theorem 9, the generalization of the Cayley-Hamilton theorem for matrices over distributive lattice with 0 and 1. In other words, Zhang [2] proved that the lattice equationis satisfied by the matrix .
Since, in the lattice case, a characteristic equation cannot be defined determinantally it is natural to choose (5) as the characteristic equation of .
Definition 10 (see [4]). Let . Then the characteristic equation of is defined as where is the symmetric group of all permutations of the set , , and , when is even; and , when is odd.
The following notations are used:
For , we have
In [4], Tan proceeds to solve (5), by assuming that the parameter in (5) is an element in . From the idempotency of and the absorption law, (5) takes the form where and , when is even; and , when is odd.
Lemma 11 (see [4]). .
Definition 12. Let and . If satisfies (10), is called a characteristic root or characteristic value of .
Theorem 13. Let and be the set of all characteristic roots of . Then is a sublattice of .
Proof. Let . Then and .
Now Therefore, , . Hence, the set of all characteristic roots of is a sublattice of .
Corollary 14. Let . Then the set of all characteristic roots of is a subset of the set of all eigenvalues of .
In 1998, Tan [4] proved the existence of the least element and the greatest element for the set of all characteristic roots of by the following theorem.
Theorem 15 (see [4]). Let . Then the set of all characteristic roots of is , where .
For simplicity, the greatest and the least characteristic roots of are denoted by and , respectively. Thus, and .
Theorem 16. Let . Then the permanent of is a characteristic root of .
Proof. We have Now Also by Lemma 11, .
Therefore, . Hence, by Theorem 15, permanent of is a characteristic root of .
Theorem 17. Let . Then is nilpotent if and only if the only characteristic root of is 0.
Proof. First assume that is nilpotent. Then by Theorem 3, . Therefore, by Theorem 15, . Hence, the only characteristic root of is 0. Conversely assume that the only characteristic root of is 0. Then by Theorem 15, . Therefore, , for . That is, . Hence, by Theorem 3, is nilpotent.
Proposition 18. Let be an idempotent matrix. Then the greatest characteristic root of is .
Proof. We have . Hence, the proof is complete.
Remark 19. Let be idempotent. Then 0 need not be a characteristic root of .
Example 20. Consider the lattice , whose diagrammatical representation is shown in Figure 1.
It is easy to see that is a distributive lattice.
Let . Then . Hence, is idempotent. Now the characteristic roots of are those values of satisfying .
Remark 21. Let . Then the characteristic roots of and need not be the same.
Example 22. In Example 20, let , . Then and . Now the characteristic roots of are those values of satisfying and the characteristic roots of are those values of satisfying .
Lemma 23 (see [9]). Let . Then and .
Theorem 24. Let . Then the characteristic roots of and are the same.
Proof. By Lemma 2(b), we have and by Lemma 23, Therefore, by Theorem 15, the characteristic roots of and are the same.
Theorem 25. Let be a triangular (upper or lower) matrix. Then the characteristic roots of are those values of satisfying
Proof. Assume that is an upper triangular matrix. Then , , .
Therefore, we have For any , there must exist some such that . Then . On the other hand, since is a term in the expression , we have .
For any with , we have and for any , and so for any . In this case, . Then Hence, By Theorem 15, the characteristic roots of are those values of satisfying If is a lower triangular matrix, then is an upper triangular matrix. Proceedings as above, we get that the characteristic roots of are those values of satisfying
Corollary 26. Let be a diagonal matrix (i.e., ). Then the characteristic roots of are those values of satisfying
Corollary 27. Let be a scalar matrix (i.e., , where ). Then “” is the only characteristic root of .
Proposition 28. Let . Then the set of all characteristic roots of is if and only if and .
Proof. First assume that the set of all characteristic roots of is .
Then by Theorem 15, we have Hence, and .
Conversely assume that and . Then the characteristic equation (10) may be written as which is true for all .
Hence, the set of all characteristic roots of is .
Proposition 29. Let . Then 0 is a characteristic root of if and only if .
Proof. First assume that 0 is a characteristic root of . Then (10) becomes Conversely assume that . Then (10) can be written as which is true when , by Lemma 11. Hence, 0 is a characteristic root of .
Example 30. Consider . We have Therefore, and .
Hence, by Proposition 28, the set of all characteristic roots of is .
Lemma 31. If is invertible, then .
Proof. Assume is invertible. Then by Theorem 4, . Using Theorems 7 and 8, we have . Hence, .
Theorem 32. Let be an invertible matrix. Then the only characteristic root of is 1.
Proof. We have Each term of is of the form , where , . Then there must exist some such that . Corresponding to that , , by Theorem 5. Hence, . Therefore, we get Similarly we can prove that .
Therefore, Hence, Therefore, .
Hence, the only characteristic root of is 1.
Definition 33. Let . If there exists an invertible matrix such that , then is said to be similar to .
Let . If there exists an orthogonal matrix such that , then is said to be orthogonally similar to .
Note that, If is similar to , then is similar to . From Theorem 4, being similar to is the same as being orthogonally similar to .
Theorem 34. Similar lattice matrices have the same characteristic roots.
Proof. Let . Assume that is similar to . Then there exists an invertible matrix such that , by Theorem 4.
We have We claim that and
To do this, we first prove for any and is invertible.
We have Similarly, we can prove and .
Therefore, Hence, Therefore, by Theorem 15, the characteristic roots of and are the same.
4. Conclusion
In this paper, we have studied the characteristic roots of different types of lattice matrices which have further strengthened the theory of lattice matrices. There is a great scope for further development in this area. We hope the research in this direction would be able to throw light on a few new concepts in lattice matrices, thereby paving the way for applications in areas ranging from switching circuits to theoretical computer science.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The second author is thankful to the University Grants Commission, New Delhi, India, for the award of Teacher Fellowship under the XII Plan period.