slope : Thus which means that the derivative measures the rate at which the successive iterates approach the fixed point or diverge from it. If the derivative at is exactly 1 or −1, then more information is needed in order to decide stability. There is an analogous criterion for a continuously differentiable map with a fixed point , expressed in terms of its Jacobian matrix at , . If all eigenvalues of are real or complex numbers with absolute value strictly less than 1 then is a stable fixed point; if at least one of them has absolute value strictly greater than 1 then is unstable. Just as for =1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is inconclusive. The same criterion holds more generally for diffeomorphisms of a smooth manifold. Linear autonomous systems The stability of fixed points of a system of constant coefficient linear differential equations of first order can be analyzed using the eigenvalues of the corresponding matrix. An autonomous system where and is an matrix with real entries, has a constant solution (In a different language, the origin is an equilibrium point of the corresponding dynamical system.) This solution is asymptotically stable as ("in the future") if and only if for all eigenvalues of , . Similarly, it is asymptotically stable as ("in the past") if and only if for all eigenvalues of , . If there exists an eigenvalue of with then the solution is unstable for . Application of this result in practice, in order to decide the stability of the origin for a linear system, is facilitated by the Routh–Hurwitz stability criterion. The eigenvalues of a matrix are the roots of its characteristic polynomial. A polynomial in one variable with real coefficients is called a Hurwitz polynomial if the real parts of all roots are strictly negative. The Routh–Hurwitz theorem implies a characterization of Hurwitz polynomials by means of an algorithm that avoids computing the roots. Non-linear autonomous systems Asymptotic stability of fixed points of a non-linear system can often be established using the Hartman–Grobman theorem. Suppose that is a -vector field in which vanishes at a point , . Then the corresponding autonomous system has a constant solution Let be the Jacobian matrix of the vector field at the point . If all eigenvalues of have strictly negative real part