Sriparna Bandopadhyay, Assistant Professor, Department of Mathematics, Indian Institute of Technology Guwahati for the award of the degree of Doctor of Philosophy and this work has not been submitted elsewhere for a degree. Manideepa Saha, student of the Department of Mathematics, Indian Institute of Technology Guwahati, for the award of the degree of Doctor of Philosophy has been carried out under my supervision and that this work has not been submitted elsewhere for a degree.
Prologue
The study of the connection between graph theory and the spectral properties of a matrix was motivated by the work of Perron and Frobenius on the spectral radius of a nonnegative matrix. Since for an arbitrary matrix M, the height and level characteristics are not equal, he questioned the nature of the relationship between them for the class of matrices M and the possible conditions under which the two characteristics are equal.
The relevance and aim of the topic of research
In [39], Noutsos generalized the class of nonnegative matrices based on the well-known Perron-Frobenius theory of this class. Based on this generalization of non-negative matrices, Elhashash and Szyld in [7], generalized the class of matrices M and called it GM-matrix.
Thesis overview
In section 2.3 we introduce the Frobenius-Jordanian form of a matrix and demonstrate the existence and uniqueness (up to similarity transformation) of such a form for non-negative matrices. Furthermore, we investigate some graph-theoretic properties of non-negative matrices using the Frobenius-Jordan form.
Notation and Preliminaries
The transitive closure of ΓpAq denoted by ΓpAq is the graph with the same vertex set as that of ΓpAqand pi, considers an edge in ΓpAqif i has access to j in ΓpAq. The reduced graph of A, denoted by RpAq is the vertex set graph consisting of all classes in A and pi, jq is an edge in RpAq if and only if it has access to j in ΓpAq.
The Frobenius Jordan Form of a Nonnegative matrix
Thus, the leading diagonal block of the Frobenius-Jordan form (and hence the Frobenius-Jordan form) of a nonnegative matrix is not unique. T Z, where Z 2I is the leading block of the Frobenius-Jordan form B, but the columns of T do not form a quasi-prime basis for A.
Nonnegative permuted graph basis for nonnegative matrices
A particularly suitable choice of T2 is achieved if the columns of T1 form a nonnegative row-switched graph basis [31] for the general space of B associated with the spectral beam of B, i.e., if of the form . For X rx1 x2 x3s all possible submatrices of X that can contribute rows to the identity matrix of the permuted graph. To obtain a criterion for the existence of nonnegative permuted graphical bases we have the following result.
Let a nonnegative matrix B partitioned as p2.2q having q basis classes have a nonnegative basis of the permuted graph for the generalized. We show that every set Ii will contribute a row (in particular, the ito row) to the identity of the basis of a nonnegative permuted graph. This contradicts the assumption that I˜j does not contribute any row to the base identity of the permuted graph.
Suppose B is a nonnegative matrix with a nonnegative permuted graph basis for the generalized eigenspace EρpBq.
Conclusion
We show that the Preferred Basis Theorem and the Index Theorem for M matrices are not true for GM matrices of order greater than 2, whereas we prove the existence of a preferred basis for a subclass of M_ matrices. We also present a procedure for obtaining a preferred basis from a quasi-preferred basis for the generalized null space of a particular subclass of M_ matrices. The existence of a quasi-preferred basis for this class of matrices was shown by Naqvi and McDonald in [33].
The existence of quasi-preferred and preferred bases for M-matrices was shown by Rothblum, Schneider and Hershkowitz in [45] and [23]. Here, using similar techniques, we give a constructive method to obtain a preferred basis from a given quasi-preferred basis for a subclass of M_ matrices. One interesting problem is to study the relation between the height and plane properties of M_-matrices.
In [33] it was proved that the height characteristic is always majorized by its level characteristic for a specific subclass of M_ matrices.
Convention
Combinatorial structure of GM-matrices
For any A P R2,2 with the spectral radius ρpAq P σpAq, the length of the longest chain of A is always smaller than or equal to indexρpAqpAq. If indexρpAqpAq 2, then the result is obviously true. indexρpAqpAq 1 and length of the longest chain2. The following example shows that the result above does not hold if the order of the matrix exceeds 2. Note that r0,1,1sT and r2,0,1sT are two linearly independent eigenvectors of A corresponding to the eigenvalue 0, thus index pAq 1.
We now give an example of a 22-matrix that satisfies the hypothesis of the above theorem and for which indexρpAqpAq is length of the longest chain in ΓpAq. In the next lemma we give a subclass of 22 matrices for which indexρpAqpAq Length of the longest chain in ΓpAq. If indexρpAqpAq is 1, then A is a diagonal matrix or has two different eigenvalues, but in both cases the length of the longest chain in ΓpAq is 1.
The following examples show that the conclusions of Theorem 3.2 and Theorem 3.3 do not hold for WP F n matrices if the order of the matrix exceeds 2.
Combinatorial structure of M _ -matrices
Preferred basis for M _ -matrices
Let A be a singular matrix and let X be such that its columns form a quasi-preferred basis of EpAq. Let us be an M-matrix andX such that the columns ofX form a quasi-preferred basis for EpAq. Let A be an M_ matrix with indexρpAq ¤ 1 and X be such that its columns form a quasi-preferred basis in EpAq.
If B is an ultimately non-negative matrix with indexpBq ¤ 1, it is known from [33] that B and hence AρIB has a quasi-preferred basis. Then we give a procedure to obtain a preferred basis based on a quasi-preferred basis for any M_ matrix A, where A ρIB with indexpBq ¤1. Suppose that
If AρIB is an M_ matrix with indexρpAq ¤1, then there is a preferred basis for EpAq.
Height and level characteristics of M _ -matrices and well
From the above lemma it is clear that for any basis of nonnegative level of EpAqand in particular for a preferred basisBofEpAq,ηpBq λpBq λpAq. Since A is an M_ matrix with indexρpAq ¤1, indexpAq is equal to the length of the longest chain in A and hence it follows that maxtlevelpykq |kP x¯tyu l. Any yi for which heightpyiq levelpyiq, heightpAkyiq heightpyiq k levelpyiq k ¥ levelpAkyiq, so it follows that heightpAkyiq levelpAkyiq for every k ¤ heightpyiq.
From the above argument it follows that if λpBq ηpBq, then there exists a yi P T with heightpyiq l such that heightpyiq nivelpyiq. ThenλipB˜q λi for all iR tl, where;λlpB˜q λl 1; λkpB˜q λk1, so it follows that λpB˜q ¡λpAq, which is a contradiction. Since every nonnegative Jordan basis for A is a nonnegative height basis for A, the 'if' part follows from Theorem 3.8 pixq.
Assume that if Ei, as defined above, do not satisfy the Hall marriage condition for all k P xt1y, then there exist k0 and α xλk0 1y such that.
Conclusion
Nonsingular M-matrices can be characterized in various ways, in terms of positivity of principal minors, stability and the inverse property of positivity (see [27]). In subsection 4.2.1 of this chapter, we give some useful characterizations of nonsingular M_-matrices in terms of positivity of sums of principal minors and stability. We have also shown that some important properties, such as inverse positivity, do not carry over to the entire class of M_-matrices, but to a subclass of these matrices.
In subsection 4.2.2 we generalize the inverse-positivity property for a subclass of nonsingular M_ matrices, given by Elhashash and Szyld in [7], to singular M_ matrices. Next, we introduce the concepts of eventual monotonicity and eventual non-negativity on a subset of Rn, to characterize a subclass of M_ matrices. In addition, this subclass of M_-matrices A with index pAq ¤ 1, is also characterized in terms of some special types of generalized inverses.
Characterization of M _ -matrices
Nonsingular M _ -matrices
Also ρ is a simple positive eigenvalue which follows from the fact that for allk¥k0,Bkis an irreducible nonnegative matrix and Lemma 3.4. Note that the above result is not true if B is nilpotent or if indexipBq ¡ 1, as the following two examples illustrate. Then B is a zero-power matrix, eventually nonnegative, and there exists no nonnegative vector corresponding to its spectral radius 0.
Since B (and thus BT) is an irreducible, non-nilpotent, eventually non-negative matrix, by Lemma 4.1ρ is simple and there exist positive vectorsx, y such that Ax λnx and yTA λnyT. Without loss of generality, we can assume that X x Xp1q. For any positive integer k, the p1,4qth and p2,3qth entries of Ak are therefore always negative, hence A1 is not eventually positive, even though B is a nonnilpotent, eventually nonnegative matrix with indexpBq 1. As EF F E 3F and F2 0, it can be easily seen that pA1qk 32kEk k3k1F, and therefore A1 is not a finally nonnegative matrix even though B is an irreducible, nonnilpotent, finally nonnegative matrix.
41] Let A sI B be an n n matrix, where B is a non-nilpotent, eventually non-negative matrix with power index k0 ¥0, i.e. k0 is the least positive integer such that Bk ¥ 0 for all k ¥ k0.
Singular M _ -matrices
The Drazin inverse of A is a matrix Y that satisfies the conditions p2q, p5q, and p6q, so it is a generalized left inverse. If AρIB is a singular M_ matrix where ρpBq ρ, B is an irreducible, nonnilpotent, finally nonnegative matrix with indexpBq ¤1, then there always exists a finally positive generalized left inverse of A. Note that we from the above statement cannot conclude that every generalized left inverse of A is ultimately positive.
The following theorem gives some characterizations of singular M-matrices in terms of generalized left inverses, obtained by Neumann and Plemmons in [35]. Note that if A is a finally nonnegative matrix such that Ak ¥0 for all k ¥g, then we can choose k0 g. We first show that if A is a matrix with indexpAq ¤ 1, then Y a generalized left inverse if and only if Y is a t1u inverse with rangepYAq rangepAq.
Now for the 'only if part' let's assume that Y is a generalized left inverse of A.
Conclusion
We use these bases to introduce a variant of the Jordan canonical form for non-negative matrices and have proven the uniqueness of such a canonical form to block triangular similarity transformation. We also study some combinatorial properties of non-negative matrices using this canonical form. Moreover, we consider permuted graph representations of non-negative bases for non-negative matrices and have derived some necessary conditions for the existence of such bases.
It is an open problem to characterize the class of nonnegative matrices that have a nonnegative permuted graph basis. We prove the existence of a preferred basis for a subclass of matrices M_ and obtain similar equivalence conditions as obtained for singular matrices M, for the equality of height and level characteristics for this class of matrices. We also try to obtain similar results for the class of GM matrices and show the existence of a favorite.
It would be interesting to study all these properties for the entire class of M_-matrices.
