On the invertibility indices and the Morse normal form for linear multivariable systems

In this paper, by extending the controllability and Brunovsky indices for a matrix pair to a matrix quadruple, three kinds of indices, say, the invertibility indices, the left invertibility indices, and the right invertibility indices, for linear time-invariant systems are introduced and studied. Based on properties of these indices, a neat expression for the Morse lists of a linear system is given without transforming it into its Morse normal form, which is a deep and fundamental result in linear systems theory, and takes an important role in many analysis and design problems for linear systems. Furthermore, the Morse normal form with only algebraic equivalence transformation is also investigated and is simplified.