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Perron–Frobenius theorem
In linear algebra, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type); to economics (Okishio's theorem, Leontief's input-output model);
to demography (Leslie population age distribution model), to Internet search engines and even ranking of football
teams.
== Statement of the Perron–Frobenius theorem ==
Let positive and non-negative respectively describe matrices with just positive real numbers as components and matrices with just non-negative real components. The eigenvalues of a real square matrix ''A'' are complex numbers that make up the spectrum of the matrix. The exponential growth rate of the matrix powers ''A''''k'' as ''k'' → ∞ is controlled by the eigenvalue of ''A'' with the largest absolute value. 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 and concerned positive matrices. Later, found their extension to certain classes of non-negative matrices.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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