Editing Perron–Frobenius theorem (section)

====Perron–Frobenius theorem for irreducible non-negative matrices====

Let <math>A</math> be an irreducible non-negative <math>N\times N</math> matrix with period <math>h</math> and [[spectral radius]] <math>\rho(A) = r</math>. 
Then the following statements hold.

* The number <math>r\in\mathbb{R}^+</math> is a positive real number and it is an eigenvalue of the matrix <math>A</math>. It is called '''Perron–Frobenius eigenvalue'''.
* The Perron–Frobenius eigenvalue <math>r</math> is [[Eigenvalues and eigenvectors#Algebraic multiplicity|simple]]. Both right and left eigenspaces associated with <math>r</math> are one-dimensional.
* <math>A</math> has both a right and a left eigenvectors, respectively <math>\mathbf v</math> and <math>\mathbf w</math>, with eigenvalue <math>r</math> and whose components are all positive. Moreover the '''only''' eigenvectors whose components are all positive are those associated with the eigenvalue <math>r</math>.
* The matrix <math>A</math> has exactly <math>h</math> (where <math>h</math> is the '''period''') complex eigenvalues with absolute value <math>r</math>. Each of them is a simple root of the characteristic polynomial and is the product of <math>r</math> with an <math>h</math>th [[root of unity]].
* Let <math>\omega = 2\pi/h</math>. Then the matrix <math>A</math> is [[similar matrix|similar]] to <math>e^{i\omega}A</math>, consequently the spectrum of <math>A</math> is invariant under multiplication by <math>e^{i\omega}</math> (i.e. to rotations of the complex plane by the angle <math>\omega</math>).
* If <math>h>1</math> then there exists a permutation matrix <math>P</math> such that
::<math>PAP^{-1}=
\begin{pmatrix}
O & A_1 & O & O & \ldots & O \\
O & O & A_2 & O & \ldots & O \\
\vdots & \vdots &\vdots & \vdots & & \vdots \\
O & O & O & O & \ldots & A_{h-1} \\
A_h & O & O & O & \ldots & O
\end{pmatrix},
</math>
::
where <math>O</math> denotes a zero matrix and the blocks along the main diagonal are square matrices.

* '''[[Lothar Collatz|Collatz]]–Wielandt formula''': for all non-negative non-zero vectors ''<math>\mathbf x
</math>'' let ''<math>f(\mathbf x)
</math>'' be the minimum value of ''<math>[A\mathbf x]_i/x_i
</math>'' taken over all those <math>i
</math> such that <math>x_i\neq0
</math>. Then <math>f
</math> is a real valued function whose [[maximum]] is the Perron–Frobenius eigenvalue.

* The Perron–Frobenius eigenvalue satisfies the inequalities
::<math>\min_i \sum_{j} a_{ij} \le r \le \max_i \sum_{j} a_{ij}.</math>

The example <math>A =\left(\begin{smallmatrix}
0 & 0 & 1 \\
0 & 0 & 1 \\
1 & 1 & 0 
\end{smallmatrix}\right)</math> shows that the (square) zero-matrices along the diagonal may be of different sizes, the blocks ''A''<sub>''j''</sub> need not be square, and ''h'' need not divide&nbsp;''n''.