Related papers: Characteristic functions and Hamilton-Cayley theor…

Motivated by the general problem of extending the classical theory of holomorphic functions of a complex variable to the case of quater- nion functions, we give a notion of an H-derivative for functions of one quaternion variable. We show…

We survey three methods for proving that the characteristic polynomial of a finite lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on…

It is a well-known fact that the first and last non-trivial coefficients of the characteristic polynomial of a linear operator are respectively its trace and its determinant. This work shows how to compute recursively all the coefficients…

In this paper a general theory of semi-classical matrix orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distributional equation $D(u A) = u B,$ where $A$ and $B$ are matrix polynomials.…

In a recent paper a class of infinite Jacobi matrices with discrete character of spectra has been introduced. With each Jacobi matrix from this class an analytic function is associated, called the characteristic function, whose zero set…

When doubly-affine matrices such as Latin and magic squares with a single non-zero eigenvalue are powered up they become constant matrices after a few steps. The process of compounding squares of orders m and n can then be used to generate…

We propose a necessary and sufficient condition for a real-valued function on the real line to be a characteristic function of a probability measures. The statement is given in terms of harmonic functions and completely monotonic functions.

The main result of the paper is the following generalization of Forelli's theorem: Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with the eigenvalues…

Left-modularity is a concept that generalizes modularity in lattice theory. In this paper, we give a characterization of left-modular elements and derive two formulae for the characteristic polynomial of a lattice with such an element, one…

In this expository paper, various properties of matrix traces, determinants and adjugate matrices are proved, including the *trace Cayley-Hamilton theorem*, which says that \[ kc_k + \sum_{i=1}^k \operatorname{Tr} (A^i) c_{k-i} = 0 \qquad…

We prove the bivariate Cayley-Hamilton theorem, a powerful generalization of the classical Cayley-Hamilton theorem. The bivariate Cayley-Hamilton theorem has three direct corollaries that are usually proved independently: The classical…

In this paper we consider the characteristic polynomial of not necessarily ranked posets. We do so by allowing the rank to be an arbitrary function from the poset to the nonnegative integers. We will prove two results showing that the…

This is a short review of some recent results obtained by the author. These results are related the problem of obtaining polynomial identities (computational formulas) for some matrix functions by means of the known polarization theorem,…

Cauchy's interlace theorem states that the characteristic polynomial of a symmetric matrix is interlaced by the characteristic polynomial of any principle submatrix. We prove this in two sentences using only the linearity of the…

Characteristic functions of linear operators are analytic functions that serve as complete unitary invariants. Such functions, as long as they are built in a natural and canonical manner, provide representations of inner functions on a…

If K/k is a function field in one variable of positive characteristic, we describe a general algorithm to factor one-variable polynomials with coefficients in K. The algorithm is flexible enough to find factors subject to additional…

The main result of the article reads: the distribution of a continuous starting from zero local martingale whose quadratic characteristic is almost surely absolutely continuous with respect to some non-random increasing continuous function…

Using the natural notion of {\em Hasse--Schmidt derivations on an exterior algebra}, we relate two classical and seemingly unrelated subjects. The first is the celebrated Cayley--Hamilton theorem of linear algebra, "{\em each endomorphism…

Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful…

In this note, the first-order Dickson polynomials are introduced through a particular case of the expression of the trace of the $n^{th}$ power of a matrix in terms of powers of the trace and determinant of the matrix itself. The technique…