Miroslav Fiedler

A note on companion matrices

Abstract We show that the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices, n −1 of them being the identity matrix in which a 2×2 identity submatrix in two consecutive rows (and columns) is replaced by an appropriate 2×2 matrix, the remaining being the identity matrix with the last entry replaced by possibly different entry. By a certain similarity transformation, we obtain a simple new companion matrix in a pentadiagonal form. Some generalizations are also possible.

A characterization of tridiagonal matrices

The purpose of this paper is to prove that symmetric irreducible tridiagonal matrices and their permutations are the only symmetric matrices (of order n > 2) the rank of which cannot be diminished to less than n - 1 by any change of diagonal elements. The main part of the proof was obtained as a byproduct of a minimum problem solution (cf. [l]).

Absolute algebraic connectivity of trees

The absolute algebraic connectivity of a graph G is the maximum of algebraic connectivities (second smallest eigenvalue of the Laplacian) for all nonnegative valuations of G whose average value on the edges of G is one. We prove that for a tree T-(V.E). this number is equal to the reciprocal of the variance of T which is the number where d(p,q) means the distance of (metric) points of the tree and M the absolute center of gravity of T, i.e. the (metric)point whose sum of squares of the distances to the vertices is minimal.(Metric means that points within the edges are also allowed). The absolute algebraic connectivity of a tree is always a rational number.

Expressing a polynomial as the characteristic polynomial of a symmetric matrix

Abstract We show that given a polynomial, one can (without knowing the roots) construct a symmetric matrix whose characteristics polynomial is the given polynomial appropriately normed. For a real polynomial, the matrix can have purely imaginary off-diagonal entries. For a polynomial with only real distinct roots, the matrix can be chosen real.

Numerical range of matrices and Levinger's theorem

Abstract This paper is, in a sense. a continuation of the authors previous paper on the numerical range of matrices. In particular, its connection to Levingers theorem for the case of nonnegative matrices is extended for general complex matrices. An elementary proof of Levingers theorem is also included.

A Geometric Approach to the Laplacian Matrix of a Graph

Let G be a finite undirected connected graph with n vertices. We assign to G an (n - 1)-simplex ∑(G) in the point Euclidean (n - 1)-space in such a way that the Laplacian L(G) of G is the Gram matrix of the outward normals of ∑(G). It is shown that the spectral properties of L(G) are reflected by the geometric shape of the Steiner circumscribed ellipsoid S of ∑(G) in a simple manner. In particular, the squares of the half-axes of S are proportional to the reciprocals of the eigenvalues of L(G). Also, a previously discovered relationship to resistive electrical circuits is mentioned.

A characterization of the Moore-Penrose inverse

Abstract If A is a nonsingular matrix of order n, the inverse of A is the unique matrix X for which rank A I I X = rank (A). We present a generalization of this fact to singular or rectangular matrices A to obtain an analogous result for the Moore-Penrose inverse A+ of A. We then give a geometric interpretation for the case that A is a positive semidefinite matrix.

Analytic functions of M-matrices and generalizations

We introduce the notion of positivity cone K of matrices in and with such a K we associate sets Z and M. For suitable choices of K the set M consists of the classical (non-singular) M-matriccs or of the positive definite (Hermitian) matrices. If AϵM and 1⩽p⩽3 we prove that there is a unique BϵM for which Bp =A. If P>3, this uniqueness theorem is false for general M and we prove a weaker result. We extend the result that for a Z-matrix A we have A −1⩾0 if and only if A is an M-matrix. Under an additional hypothesis on the positivity cone, we exhibit a class of entire functions f(z) such that for A ϵ Z we have A ϵ M if and only if there is a B ϵ K for which f(B) = A −1.

A classification of matrices of class Z

Abstract We generalize the classes N 0 and F 0 studied by K. Fan, G. Johnson, and R. Smith. Schur complements and lattices are examined for matrices in these classes.

Rank-preserving diagonal completions of a matrix

Abstract Suppose A is an n-by-n matrix over a field F. We prove that it is possible to complete the diagonal entries of A so that the resulting rank of A is as small as possible when n⩾3r, where r is the “off-diagonal rank” of A and (n,r)≠(3,1).