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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 A λ λ = − B 0 . There are other applications of working directly with the characteristic polynomial, for instance, using Routh-Hurwitz analysis to count the stable roots of 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 and a lexicographic combination of the rows between matrices A and B. ( ) P λ