A rational canonical form for matrix fields

A Rational Canonical Form Algorithm

Rational Canonical Form

In mathematics, complete classification of structures, such as groups and rings, is often a primary goal. Linear transformations are no exception to this. Certain canonical forms exist to classify linear transformations, therefore creating a unique representative of linear transformations in the same similarity class. Diagonal representation is of course one of the simplest examples of a canoni...

A Markovian canonical form of second-order matrix-exponential processes

Besides the fact that – by definition – matrix-exponential processes (MEPs) are more general than Markovian arrival processes (MAPs), only very little is known about the precise relationship of these processes in matrix notation. For the first time, this paper proves the persistent conjecture that – in two dimensions – the respective sets, MAP(2) and MEP(2), are indeed identical with respect to...

Computing the Combinatorial Canonical Form of a Layered Mixed Matrix

This paper presents an improved algorithm for computing the Combinatorial Canonical Form (CCF) of a layered mixed matrix A = Q T , which consists of a numerical matrix Q and a generic matrix T . The CCF is the (combinatorially unique) nest block-triangular form obtained by the row operations on the Q-part, followed by permutations of rows and columns of the whole matrix. The main ingredient of ...

A Finiteness Theorem for Canonical Heights Attached to Rational Maps over Function Fields

Let K be a function field, let φ ∈ K(T ) be a rational map of degree d ≥ 2 defined over K, and suppose that φ is not isotrivial. In this paper, we show that a point P ∈ P(K̄) has φ-canonical height zero if and only if P is preperiodic for φ. This answers affirmatively a question of Szpiro and Tucker, and generalizes a recent result of Benedetto from polynomials to rational functions. We actually...