R.B. Bapat

A max version of the Perron-Frobenius theorem

Abstract If A an n × n nonnegative, irreducible matrix, then there exists μ(A) > 0, and a positive vector x such that maxjaijxj = μ(A)xi, i = 1, 2,…, n. Furthermore, μ(A) is the maximum geometric mean of a circuit in the weighted directed graph corresponding to A. This theorem, which we refer to as the max version of the Perron-Frobenius Theorem, is well-known in the context of matrices over the max algebra and also in the context of matrix scalings. In the present work, which is partly expository, we bring out the intimate connection between this result and the Perron-Frobenius theory. We present several proofs of the result, some of which use the Perron-Frobenius Theorem. Structure of max eigenvalues and max eigenvectors is described. Possible ways to unify the Perron-Frobenius Theorem and its max version are indicated. Some inequalities for μ(A) are proved.

Generalized inverses over integral domains

In an earlier paper one of the authors showed that a matrix of rank r over an integral domain has a generalized inverse if and only if a linear combination of all the r × r minors of the matrix is one. In the same paper a procedure for constructing a generalized inverse from such a linear combination was also given. In the present paper we show that any reflexive generalized inverse can be obtained by that procedure. We also show that if the integral domain is a principal ideal domain, every generalized inverse can be obtained by that procedure. It is also shown that a matrix A of rank r over an integral domain has Moore-Penrose inverse if and only if the sum of squares of all r × r minors of A is an invertible element of the integral domain.

Generalized matrix tree theorem for mixed graphs

In this article we provide a combinatorial description of an arbitrary minor of the Laplacian matrix (L) of a mixed graph (a graph with some oriented and some unoriented edges). This is a generalized Matrix Tree Theorem. We also characterize the non-singular substructures of a mixed graph. The sign attached to a nonsingular substructure is described in terms of labeling and the number of unoriented edges included in certain paths. Nonsingular substructures may be viewed as generalized matchings, because in the case of disjoint vertex sets corresponding to the rows and columns of a minor of L, our generalized Matrix Tree Theorem provides a signed count over matchings between those vertex sets. A mixed graph is called quasi bipartite if it does not contain a non singular cycle (a cycle containing an odd number of un-oriented edges). We give several characterizations of quasi-bipartite graphs.

Algebraic connectivity and the characteristic set of a graph

Let Gbe a connected weighted graph on vertices {1,2,…,n} and L be the Laplacian matrix of GLet μ be the second smallest eigenvalue of L and Y be an eigenvector corresponding to μ. A characteristic vertex is a vertex v such that Y(v) = 0 and Y(w) ≠ 0 for some vertex w adjacent to v. An edge e with end vertices v,w is called a characteristic edge of G if Y(w) Y(v) < 0. The characteristic vertices and the characteristic edges together form the characteristic set of G. We investigate the characteristic set of an arbitrary graph. The relation between the characteristic set and nonnegative matrix theory is exploited. Bounds are obtained on the cardinality of the characteristic set. It is shown that if G is a connected graph with n vertices and m edges then the characteristic set has at most m − n + 2 elements. We use the description of the Moore-Penrose inverse of the vertex-edge incidence matrix of a tree to derive a classical result of Fiedler for a tree. Furthermore, an analogous result is obtained for an ei...

On majorization and Schur products

Abstract Suppose A, D1,…,Dm are n × n matrices where A is self-adjoint, and let X = Σ m k = 1 D k AD ∗ k . It is shown that if ΣD k D ∗ k = ΣD ∗ k D k = I , then the spectrum of X is majorized by the spectrum of A. In general, without assuming any condition on D1,…,Dm, a result is obtained in terms of weak majorization. If each Dk is a diagonal matrix, then X is equal to the Schur (entrywise) product of A with a positive semidefinite matrix. Thus the results are applicable to spectra of Schur products of positive semidefinite matrices. If A, B are self-adjoint with B positive semidefinite and if bii = 1 for each i, it follows that the spectrum of the Schur product of A and B is majorized by that of A. A stronger version of a conjecture due to Marshall and Olkin is also proved.

Mixed discriminants of positive semidefinite matrices

Abstract If Ak=(akij), k= 1,2,…,n, are n-by-n matrices, then their mixed discriminant D(A1,…,An) is given by D(A 1 ,…,A n = 1 n! ∑ σϵS n a σ(j) ij , where Sn is the symmetric group of degree n and where |·| denotes determinant. We give certain alternative ways of defining the mixed discriminant and state some basic properties. It is pointed out that a Ryser-type formula for the mixed discriminant exists in the literature, and a simpler proof is given for it. It is shown that the mixed discriminant can be expressed as an inner product. A generalization of Konigs theorem on 0–1 matrices is proved. The following set D n, which includes the set of n-by-n doubly stochastic matrices, is defined and studied: .

On likelihood-ratio ordering of order statistics

Let X1,…, Xn be independent random variables such that X1<lr X2<lr … <lr Xn , where <lr denotes likelihood-ratio ordering. It is shown that, under mild assumptions, the corresponding order statistics X1 ⩽ … ⩽ X(n) are similarly ordered, i.e., X(1)<lr X(2)<lr … <lr X(n) .

Pattern Properties and Spectral Inequalities in Max Algebra

The max algebra consists of the set of real numbers, along with negative infinity, equipped with two binary operations, maximization and addition. This algebra is useful in describing certain conventionally nonlinear systems in a linear fashion. Properties of eigenvalues and eigenvectors over the max algebra that depend solely on the pattern of finite and infinite entries in the matrix are studied. Inequalities for the maximal eigenvalue of a matrix over the max algebra, motivated by those for the Perron root of a nonnegative matrix, are proved.

Asymptotics of the Perron eigenvalue and eigenvector using max-algebra

Abstract We consider the asymptotics of the Perron eigenvalue and eigenvector of irreducible nonnegative matrices whose entries have a geometric dependance in a large parameter. The first term of the asymptotic expansion of these spectral elements is solution of a spectral problem in a semifield of jets, which generalizes the max-algebra. We state a “Perron-Frobenius theorem” in this semifield, which allows us to characterize the first term of this expansion in some non-singular cases. The general case involves an aggregation procedure a la Wentzell-Freidlin.

The Generalized Moore-Penrose Inverse

Abstract We define the generalized Moore-Penrose inverse and give necessary and sufficient conditions for its existence over an integral domain. We also prove its uniqueness and give a formula for it which leads us towards a “generalized Cramers rule” to find the generalized Moore-Penrose solution.