A New, Necessary and Sufficient condition for Hurwitz Stability of a Real Matrix Without Characteristic Polynomial, Using Qualitative Reasoning

In this paper, we present a new, necessary and sufficient condition for Hurwitz stability of real matrix, using qualitative (based solely on sign pattern of the matrix) reasoning. This new stability condition, completely deviates from the age-old quantitative methods such as the Routh-Hurwitz criterion, Lyapunov Matrix Solution criterion and the related Fuller's criterion, which are all constrained by the similarity transformation phenomenon. Put it another way, this new condition does not need the formation of the characteristic polynomial at all and is based on the direct matrix entries' information. We propose a new, necessary and sufficient condition for the Hurwitz stability of any real matrix, in terms of three `critical matrix indices' each for the original matrix A and its higher dimensional Bialternate sum matrix (labeled Fuller's matrix), denoted by \mathscrB. Thus the new necessary and sufficient condition relies on the behavior of a total six `critical matrix indices'. It can be said that through the condition presented in this paper, we are assessing the Hurwitz stability of a real matrix without using the Routh-Hurwitz criterion. Examples are given to illustrate the proposed methodology along with a discussion of the attractive `convexity' property of this new condition which is lacking in the current quantitative methods.