Frobenius, theorem

This Hamiltonian has only non-positive matrix elements and its matrix in the space of spin configurations cannot be represented in a block-diagonal form after any permutation of the basis functions. Therefore, according to the Perron-Frobenius theorem its ground state must be nondegenerate. Obviously, the... 

The problem is to determine the matrix T in such a way that the new variable q = co1(a , y, z) retains some biological interpretation while at the same time the new system becomes more tractable. The next result describes how to do this it is based on the Perron-Frobenius theorem (see Appendix A, Theorem A.4). 

We now see that parts 1 and 2 of the Coulson-Rushbrooke Theorem are really a special case of the matrix theorem proved in D1 in fact, this matrix theorem is, in its turn, only a special case of the famous Perron-Frobenius Theorem on matrices with non-negative elements (1907-1912)R4T. 

Some induction schemes for ct, n, and 5 orbital basis sets on C y sites of polyhedral complexes are to be found in Appendix D. In addition to the Frobenius theorem, there is also a stronger result for induction theory based on the concept of a fiber bundle. This requires the coupling of representations and will be considered in Sect. 6.9. 

The site symmetry of a cube is T. The cube is an invariant of its site group and transforms as ag in T. The set of five cubes thus spans the induced representation aTh /ft. Applying the Frobenius theorem to the subduction (see Sect. C.l), one obtains... 

Facts (a)-(c) are a generalized Perron- Frobenius theorem fact (d) is a consequence of a generalized spectral radius formula. Note that the worst-case rate of eonvergence to equilibrium from an initial nonequilibriiun distribution is eon trolled by R, and hence by Texp. 

Theorem A. 14 (Perron-Frobenius Theorem). If A is a nonnegative irreducible matrix then A possesses a unique eigenvalue A = A (A), called the Perron-Frobenius eigenvalue, such that... 

From this formula and the Perron-Frobenius Theorem one easily infers that if A is irreducible and B is a non-negative matrix with Bij > 0 for at least a choice of i and j, then A (A - - B) > A (A). Since A is simple... 

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