Perron-Frobenius theory

Proof. Assertion (1) is just Theorem C.4. The assertion concerning M (xf) follows from the Perron-Frobenius theory (Theorem A.5) and the monotonicity of the time-reversed system (6.3). If J is the Jacobian matrix of / at Xq, then (6.2) implies that —J satisfies the hypotheses of Theorem A.5. It follows that r = —s —J) < 0 is an eigenvalue of J corresponding to an eigenvector u > 0. Because M (Xo) is tangent at Xq to the line through Xq in the direction v, the local stable manifold of Xq is totally ordered. Since M X()) is the extension of the local stable manifold by the order-preserving backward (or time-reversed) system, it follows... 

For special matrices there are theorems that give information about the stability modulus. A matrix is said to be positive if all of the entries are positive this is written A > 0. (Similarly, a matrix is nonnegative if all of the entries are nonnegative.) The very elegant Perron-Frobenius theory applies to such matrices. 

The theoretical results for discrete approximations to (2.1) and (2.6) are strongly dependent on the Perron-Frobenius theory [28 14 50 7] of nonnegative matrices, and suitable extensions thereof. It is interesting, but not surprising, that the practical numerical methods which have been used to solve the discrete approximations to (2.1) and (2.6) are also greatly influenced by this same theory of non-negative matrices. In later sections, we shall give specific numerical applications which result from this theory. 

Since non-negative square matrices leave the positive hyperoctant invariant, the application of the Perron-Frobenius theory of non-negative matrices in the discrete case is closely related to the abstractions in the papers in this same volume by G. Birkhoff, and G. Habetler and M. A. Martino. 

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

Such matrices are called stochastic matrices ) and have been studied by Perron and Frobenius. It is clear that T has a left eigenvector (1,1,..., 1) with eigenvalue 1 and therefore a right eigenvector ps such that Tps = ps, which is the Pi(y) of the stationary process. It is not necessarily a physical equilibrium state, but may, e.g., represent a steady state in which a constant flow is maintained. The principal task of the theory is to show that for any initial p(0)... 

Work of Perron and Frobenius. It was shown in [3] that the concepts of criticality, multiplication factor, period, and importance function could be deduced in finite models from the theories of positive and non-negative matrices, developed a half-century ago by Perron and Frobenius. From the heuristic assumption that an arbitrarily close approximation to the behavior of any reactor can be obtained, by dividing the neutron phase-space 2 into sufficiently small cells , one would expect the behavior of continuous reactors to follow similar lines. As explained in [3], digital computations can also be interpreted as referring to finite-dimensional multiplicative processes, of the type (7)-(7a) or (8)-(8a). It therefore seems appropriate... 

This is the usual definition of positivity in the theory of vector lattices it is weaker than the definition of Perron and Frobenius, according to which a vector is positive (in symbols, / > 0) when all its components are positive. In lattice theory, such vectors are called weak units . 

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