Equivalence of the Hurwitz and Routh criteria

The validity of the Routh criterion can be verified easily by the equivalence with the Hurwitz criterion. From the coefficients of the first row of the Routh schema the connection with the Hurwitz determinants can be seen directly:

The coefficients in the first column of the Routh schema are just the quotients of consecutive Hurwitz determinants. When all Hurwitz determinants are positive, then their quotients and therefore also the coefficients in the first column of the Routh schema are positive. When the coefficients of the Routh schema are positive, then also, as , all Hurwitz determinants are positive. Thus, the Routh criterion is equivalent to the Hurwitz criterion.