On the Characteristic Polynomial of Regular Linear Matrix Pencil

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 ( A,B ) is by way of its characteristic polynomial, 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 A and B .