It is well known that positive definiteness of a suitable Liapunov fun
ction is equivalent to the Hurwitz-Routh test. This paper uses the sta
te covariance matrix in a quadratic Liapunov function to prove the Hur
witz-Routh test and to show a conservation principle relating stabilit
y (the characteristic coefficients) to L(2) performance. Our proof req
uires the positive definite test of two matrices approximately half th
e size of the state, a savings over the usual positive definite test o
f an n x n Liapunov matrix.