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Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.
In general, any measurable function can be pushed forward. The push-forward then becomes a linear operator, known as the transfer operator or Frobenius–Perron operator.In finite spaces this operator typically satisfies the requirements of the Frobenius–Perron theorem, and the maximal eigenvalue of the operator corresponds to the invariant measure.
The technical details may be found in "A spectral theoretic proof of Perron-Frobenius", Mathematical Proceedings of the Royal Irish Academy, 102A (1), 29-35 (2002). I described the canonical form as "folklore" because although it's essential for the non-negative application I've never seen it properly attributed to anyone.
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
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 .
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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.