## Abstract

Linear matrix pencil, denoted by ( * A,B * ), plays an important role in control systems and numerical linear algebra. The problem of finding the eigenvalues of (A,B) is often solved numerically by using the well-known QZ method. Another approach for exploring the eigenvalues of (

*) is by way of its characteristic polynomial,*

**A,B***P*(λ)=

*A*− λ

*B*. There are other applications of working directly with the characteristic polynomial, for instance, using Routh-Hurwitz analysis to count the stable roots of

*P*(λ) and transfer function representation of control systems governed by differential-algebraic equations. In this paper, we present an algorithm for algebraic construction of the characteristic polynomial of a regular linear pencil. The main theorem reveals a connection between the coefficients of

*P*(λ) and a lexicographic combination of the rows between matrices

*and*

**A***.*

**B**Original language | American English |
---|---|

Journal | Alexandria Journal of Mathematics |

Volume | 1 |

State | Published - Jun 1 2010 |

## Keywords

- Characteristic polynomial
- Combinatorics
- Generalized eigenvalue problem
- Regular matrix pencil
- choose function
- lexicographic order

## DC Disciplines

- Education
- Mathematics