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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.
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 .
The exponential of a matrix A is defined by =!. Given a matrix B, another matrix A is said to be a matrix logarithm of B if e A = B.. Because the exponential function is not bijective for complex numbers (e.g. = =), numbers can have multiple complex logarithms, and as a consequence of this, some matrices may have more than one logarithm, as explained below.
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.
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
A Frobenius matrix is a special kind of square matrix from numerical analysis. A matrix is a Frobenius matrix if it has the following three properties: all entries on the main diagonal are ones; the entries below the main diagonal of at most one column are arbitrary; every other entry is zero; The following matrix is an example.
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.
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) Pitman–Koopman–Darmois theorem