The number $r\in \mathbb {R} ^{+}$ is a positive real number and it is an eigenvalue of the matrix $A$ . It is called Perron–Frobenius eigenvalue.

The Perron–Frobenius eigenvalue $r$ is . Both right and left eigenspaces associated with $r$ are one-dimensional.

simple

$A$ has both a right and a left eigenvectors, respectively $\mathbf {v}$ and $\mathbf {w}$ , with eigenvalue $r$ and whose components are all positive. Moreover these are the only eigenvectors whose components are all positive are those associated with the eigenvalue $r$ .

The matrix $A$ has exactly $h$ (where $h$ is the period) complex eigenvalues with absolute value $r$ . Each of them is a simple root of the characteristic polynomial and is the product of $r$ with an $h$ th .

root of unity

Let $\omega =2\pi /h$ . Then the matrix $A$ is to $e^{i\omega }A$ , consequently the spectrum of $A$ is invariant under multiplication by $e^{i\omega }$ (i.e. to rotations of the complex plane by the angle $\omega$ ).

similar

If $h>1$ then there exists a permutation matrix $P$ such that

Counterexamples[edit]

The matrices L = $\left({\begin{smallmatrix}1&0&0\\1&0&0\\1&1&1\end{smallmatrix}}\right)$ , P = $\left({\begin{smallmatrix}1&0&0\\1&0&0\\\!\!\!-1&1&1\end{smallmatrix}}\right)$ , T = $\left({\begin{smallmatrix}0&1&1\\1&0&1\\1&1&0\end{smallmatrix}}\right)$ , M = $\left({\begin{smallmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&1&0&0\end{smallmatrix}}\right)$ provide simple examples of what can go wrong if the necessary conditions are not met. It is easily seen that the Perron and peripheral projections of L are both equal to P, thus when the original matrix is reducible the projections may lose non-negativity and there is no chance of expressing them as limits of its powers. The matrix T is an example of a primitive matrix with zero diagonal. If the diagonal of an irreducible non-negative square matrix is non-zero then the matrix must be primitive but this example demonstrates that the converse is false. M is an example of a matrix with several missing spectral teeth. If ω = e^iπ/3 then ω⁶ = 1 and the eigenvalues of M are {1,ω²,ω³=-1,ω⁴} with a dimension 2 eigenspace for +1 so ω and ω⁵ are both absent. More precisely, since M is block-diagonal cyclic, then the eigenvalues are {1,-1} for the first block, and {1,ω²,ω⁴} for the lower one

Terminology[edit]

A problem that causes confusion is a lack of standardisation in the definitions. For example, some authors use the terms strictly positive and positive to mean > 0 and ≥ 0 respectively. In this article positive means > 0 and non-negative means ≥ 0. Another vexed area concerns decomposability and reducibility: irreducible is an overloaded term. For avoidance of doubt a non-zero non-negative square matrix A such that 1 + A is primitive is sometimes said to be connected. Then irreducible non-negative square matrices and connected matrices are synonymous.^[33]

The nonnegative eigenvector is often normalized so that the sum of its components is equal to unity; in this case, the eigenvector is the vector of a probability distribution and is sometimes called a stochastic eigenvector.

Perron–Frobenius eigenvalue and dominant eigenvalue are alternative names for the Perron root. Spectral projections are also known as spectral projectors and spectral idempotents. The period is sometimes referred to as the index of imprimitivity or the order of cyclicity.

– Variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces

Min-max theorem

– Square matrix whose off-diagonal entries are nonpositive

Z-matrix (mathematics)

M-matrix

– Complex square matrix for which every principal minor is positive

P-matrix

– Algebraic matrix element to analyze a polynomial by its coefficients

Hurwitz matrix

(Quasipositive matrix)

Metzler matrix

– self-adjoint (or Hermitian) element A of a C*-algebra A is called positive if its spectrum σ (A) consists of non-negative real numbers

Positive operator

– A generalization of the Perron–Frobenius theorem to Banach spaces

Krein–Rutman theorem

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