Assuming no roots of lie on the imaginary axis, and letting
= The number of roots of with negative real parts, and
= The number of roots of with positive real parts
then we have
Expressing in polar form, we have
from (2) note that
Now if the ith root of has a positive real part, then (using the notation y=(RE[y],IM[y]))
Similarly, if the ith root of has a negative real part,
Therefore, when the ith root of has a positive real part, and when the ith root of has a negative real part. Alternatively,
So, if we define
then we have the relationship
and combining (3) and (16) gives us
Therefore, given an equation of of degree we need only evaluate this function to determine , the number of roots with negative real parts and , the number of roots with positive real parts.
Equations (13) and (14) show that at , is an integer multiple of . Note now, in accordance with (6) and Figure 1, the graph of vs , that varying over an interval (a,b) where and are integer multiples of , this variation causing the function to have increased by , indicates that in the course of travelling from point a to point b, has "jumped" from to one more time than it has jumped from to . Similarly, if we vary over an interval (a,b) this variation causing to have decreased by , where again is a multiple of at both and , implies that has jumped from to one more time than it has jumped from to as was varied over the said interval.
Thus, is times the difference between the number of points at which jumps from to and the number of points at which jumps from to as ranges over the interval provided that at , is defined.
In the case where the starting point is on an incongruity (i.e. , i = 0, 1, 2, ...) the ending point will be on an incongruity as well, by equation (16) (since is an integer and is an integer, will be an integer). In this case, we can achieve this same index (difference in positive and negative jumps) by shifting the axes of the tangent function by , through adding to . Thus, our index is now fully defined for any combination of coefficients in by evaluating over the interval (a,b) = when our starting (and thus ending) point is not an incongruity, and by evaluating
over said interval when our starting point is at an incongruity.
This difference, , of negative and positive jumping incongruities encountered while traversing from to is called the Cauchy Index of the tangent of the phase angle, the phase angle being or , depending as is an integer multiple of or not.
To derive Routh's criterion, first we'll use a different notation to differentiate between the even and odd terms of :
Now we have:
Therefore, if is even,
and if is odd:
Now observe that if is an odd integer, then by (3) is odd. If is an odd integer, then is odd as well. Similarly, this same argument shows that when is even, will be even. Equation (13) shows that if is even, is an integer multiple of . Therefore, is defined for even, and is thus the proper index to use when n is even, and similarly is defined for odd, making it the proper index in this latter case.
Thus, from (6) and (22), for even:
and from (18) and (23), for odd:
Lo and behold we are evaluating the same Cauchy index for both:
Sturm gives us a method for evaluating . His theorem states as follows:
Given a sequence of polynomials where:
1) If then , , and
and we define as the number of changes of sign in the sequence for a fixed value of , then:
A sequence satisfying these requirements is obtained using the Euclidean algorithm, which is as follows:
Starting with and , and denoting the remainder of by and similarly denoting the remainder of by , and so on, we obtain the relationships:
or in general
where the last non-zero remainder, will therefore be the highest common factor of . It can be observed that the sequence so constructed will satisfy the conditions of Sturm's theorem, and thus an algorithm for determining the stated index has been developed.
It is in applying Sturm's theorem (28) to (26), through the use of the Euclidean algorithm above that the Routh matrix is formed.
and identifying the coefficients of this remainder by , , , , and so forth, makes our formed remainder
Continuing with the Euclidean algorithm on these new coefficients gives us
where we again denote the coefficients of the remainder by , , , ,
making our formed remainder
and giving us
The rows of the Routh array are determined exactly by this algorithm when applied to the coefficients of (19). An observation worthy of note is that in the regular case the polynomials and have as the highest common factor and thus there will be polynomials in the chain .
Note now, that in determining the signs of the members of the sequence of polynomials that at the dominating power of will be the first term of each of these polynomials, and thus only these coefficients corresponding to the highest powers of in , and , which are , , , , ... determine the signs of , , ..., at .
So we get that is, is the number of changes of sign in the sequence , , , ... which is the number of sign changes in the sequence , , , , ... and ; that is is the number of changes of sign in the sequence , , , ... which is the number of sign changes in the sequence , , , , ...
Since our chain , , , , ... will have members it is clear that since within if going from to a sign change has not occurred, within going from to one has, and likewise for all transitions (there will be no terms equal to zero) giving us total sign changes.
As and , and from (17) , we have that and have derived Routh's theorem -
The number of roots of a real polynomial which lie in the right half plane is equal to the number of changes of sign in the first column of the Routh scheme.
And for the stable case where then by which we have Routh's famous criterion:
In order for all the roots of the polynomial to have negative real parts, it is necessary and sufficient that all of the elements in the first column of the Routh scheme be different from zero and of the same sign.
Hurwitz, A., "On the Conditions under which an Equation has only Roots with Negative Real Parts", Rpt. in Selected Papers on Mathematical Trends in Control Theory, Ed. R. T. Ballman et al. New York: Dover 1964
Routh, E. J., A Treatise on the Stability of a Given State of Motion. London: Macmillan, 1877. Rpt. in Stability of Motion, Ed. A. T. Fuller. London: Taylor & Francis, 1975
Felix Gantmacher (J.L. Brenner translator) (1959) Applications of the Theory of Matrices, pp 177–80, New York: Interscience.