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Analysis of Hurwitz stability/instability of a real matrix via the concepts of Qualitative Determinant and Signature of a matrix

Currently available conditions of Hurwitz stability/instability assessment of a real matrix are essentially based on quantitative information (both sign and magnitudes of the entries of the matrix). Assessing the Hurwitz stability/instability, solely based on sign information, labeled Qualitative (Sign) Stability/Instability is also of much importance as a supplement to these quantitative based results. In this paper, we highlight the importance of the `elemental sign structure' of a matrix in its Hurwitz stability/instability assessment and present new results in the form of necessary and sufficient conditions. This analysis is done using the concepts of both Quantitative Determinant (involving magnitudes) and the Qualitative Determinant (involving only the sign information of the matrix elements). Using these metrics, we form the `Signature' of a real matrix that reflects the stability/instability nature of the matrix. The proposed results in this paper are deemed helpful also in solving many other related problems of stability including the testing of robust stability of interval parameter matrix families, which has attracted intense attention and scrutiny in the last few decades.