Simple proof of the Routh stability criterion based on order reduction of polynomials and principle of argument

Simple proof for the Routh stability criterion is derived by applying the principle of the argument to the explicit polynomial order reduction formula which generates each row of the Routh table. By this proof, we can easily reach the famous result that the number of times of sign changes in the first column elements in the Routh table coincides with the number of unstable zeros of a given polynomial. Our proof is easily extendible to the stability tests for complex coefficients systems and time-delay systems