Search results
Results from the WOW.Com Content Network
The Perron–Frobenius theorem describes the properties of the leading eigenvalue and of the corresponding eigenvectors when A is a non-negative real square matrix. Early results were due to Oskar Perron ( 1907 ) and concerned positive matrices.
The left-adjoint of the Perron–Frobenius operator is the Koopman operator or composition operator. The general setting is provided by the Borel functional calculus . As a general rule, the transfer operator can usually be interpreted as a (left-) shift operator acting on a shift space .
Frobenius reciprocity theorem in group representation theory describing the reciprocity relation between restricted and induced representations on a subgroup Perron–Frobenius theorem in matrix theory concerning the eigenvalues and eigenvectors of a matrix with positive real coefficients
On the other hand, the Perron–Frobenius theorem also ensures that every irreducible stochastic matrix has such a stationary vector, and that the largest absolute value of an eigenvalue is always 1. However, this theorem cannot be applied directly to such matrices because they need not be irreducible. In general, there may be several such vectors.
Perron numbers are named after Oskar Perron; the Perron–Frobenius theorem asserts that, for a real square matrix with positive algebraic entries whose largest eigenvalue is greater than one, this eigenvalue is a Perron number. As a closely related case, the Perron number of a graph is defined to be the spectral radius of its adjacency matrix.
A consequence (requiring additional arguments) of the above theorem is the following: [1] If φ ∈ Out(F k) is irreducible then the Perron–Frobenius eigenvalue λ(f) does not depend on the choice of a train track representative f of φ but is uniquely determined by φ itself and is denoted by λ(φ).
For example if T is a positive matrix with spectral radius r then the Perron–Frobenius theorem asserts that r ∈ σ(T). The associated spectral projection P = P ( r ; T ) is also positive and by mutual orthogonality no other spectral projection can have a positive row or column.
The Frobenius theorem can be restated more economically in modern language. Frobenius' original version of the theorem was stated in terms of Pfaffian systems, which today can be translated into the language of differential forms. An alternative formulation, which is somewhat more intuitive, uses vector fields.