Simultaneous Similarity Reductions for a Pair of Matrices to Condensed Forms

We present simultaneous reduction algorithms for two(nonsymmetric)matrices A and B to upper Hessenberg and lower Hessenberg forms,respectively.One is through the simultaneous similarity reduction and the other is through a Lanczos–Arnoldi-type iteration.The algorithm that uses the Lanczos–Arnoldi-type iteration can be considered as a generalization of both the nonsymmetric Lanczos algorithm and the standard Arnoldi algorithm.We shall also apply our reduction to construct a model reduction for certain kind second-order single-input single-output system.It is proved that the model reduction has the desirable moment matching property.