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On the final limit of a transition matrix

Helmut Kahl

Abstract

An efficient algorithm is presented for computation of the limit of exp(t A) for t towards infinity where A denotes an intensity matrix of finite dimension.

On the final limit of a transition matrix

Abstract

An efficient algorithm is presented for computation of the limit of exp(t A) for t towards infinity where A denotes an intensity matrix of finite dimension.

Paper Structure

This paper contains 4 sections, 8 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. The final limit of a stochastic matrix
  3. The final limit of an intensity matrix
  4. Computation of the final limit

Key Result

Theorem 1

The final limit $P$ of $A \in \mathbb{C}^{m \times m}$ exists if and only if $A$ is semistable. If $P$ exists $\mathbb{C}^m$ is the direct sum of $N(A)$ and $R(A)$, and it holds $P A = A P = O$. Hence $x \mapsto x P$ is the projector onto $N(A)$ along $R(A)$.

Theorems & Definitions (15)

  • Theorem 1
  • Proposition 2
  • Corollary 3
  • proof
  • Corollary 4
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • Remark 7
  • ...and 5 more