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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.
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.
Perpendicular axis theorem ; Perron–Frobenius theorem (matrix theory) Peter–Weyl theorem (representation theory) Phragmén–Lindelöf theorem (complex analysis) Picard theorem (complex analysis) Picard–Lindelöf theorem (ordinary differential equations) Pick's theorem ; Pickands–Balkema–de Haan theorem (extreme value theory)
Oskar Perron (7 May 1880 – 22 February 1975) was a German mathematician. He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951. He made numerous contributions to differential equations and partial differential equations , including the Perron method to solve the Dirichlet ...
In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, ...
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 .
In functional analysis, the Krein–Rutman theorem is a generalisation of the Perron–Frobenius theorem to infinite-dimensional Banach spaces. [1] It was proved by Krein and Rutman in 1948. [ 2 ]
In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of inequivalent definitions of the integral of a function.