Editing Perron–Frobenius theorem (section)

====Classification of matrices====
Let ''A'' be a  ''n''&nbsp;×&nbsp;''n'' square matrix over [[field (mathematics)|field]] ''F''.
The matrix ''A'' is '''irreducible''' if any of the following equivalent properties
holds.

'''Definition 1 :''' ''A'' does not have non-trivial invariant ''coordinate'' subspaces.
Here a non-trivial coordinate subspace means a [[linear subspace]] spanned by any [[proper subset]] of standard basis vectors of ''F<sup>n</sup>''. More explicitly, for any linear subspace spanned by standard basis vectors ''e''<sub>''i''<sub>1</sub> </sub>, ...,
''e''<sub>''i''<sub>k</sub></sub>, 0&nbsp;< ''k''&nbsp;<&nbsp;''n'' its image under the action of ''A'' is not contained in the same subspace.

'''Definition 2:''' ''A'' cannot be conjugated into block upper triangular form by a [[permutation matrix]] ''P'':
: <math>PAP^{-1} \ne
\begin{pmatrix} E & F \\ O & G \end{pmatrix},</math>
where ''E'' and ''G'' are non-trivial (i.e. of size greater than zero) square matrices.

'''Definition 3:''' One can associate with a matrix ''A'' a certain [[directed graph]] ''G''<sub>''A''</sub>. It has ''n'' vertices labeled 1,...,''n'', and there is an edge from vertex ''i'' to vertex ''j'' precisely when ''a''<sub>''ij''</sub> ≠ 0. Then the matrix ''A'' is irreducible if and only if its associated graph ''G''<sub>''A''</sub> is [[strongly connected component|strongly connected]].

If ''F'' is the field of real or complex numbers, then we also have the following condition.

'''Definition 4:''' The [[group representation]] of <math>(\mathbb R, +)</math> on <math>\mathbb{R}^n</math> or <math>(\mathbb C, +)</math> on <math>\mathbb{C}^n</math> given by <math>t \mapsto\exp(tA)</math> has no non-trivial invariant coordinate subspaces. (By comparison, this would be an [[irreducible representation]] if there were no non-trivial invariant subspaces at all, not only considering coordinate subspaces.)

A matrix is '''reducible''' if it is not irreducible.

A real matrix ''A'' is '''primitive''' if it is non-negative and its ''m''th power is positive for some natural number ''m'' (i.e. all entries of ''A<sup>m</sup>'' are positive).

Let ''A'' be real and non-negative. Fix an index ''i'' and define the '''period of index ''i'' ''' to be the [[greatest common divisor]] of all natural numbers ''m'' such that (''A''<sup>''m''</sup>)<sub>''ii''</sub> > 0. When ''A'' is irreducible, the period of every index is the same and is called the '''period of ''A''. ''' In fact, when ''A'' is irreducible, the period can be defined as the greatest common divisor of the lengths of the closed directed paths in ''G''<sub>''A''</sub> (see Kitchens<ref name="Kitchens"/> page 16). The period is also called the index of imprimitivity (Meyer<ref name="Meyer"/> page 674) or the order of cyclicity. If the period is 1, ''A'' is '''aperiodic'''. It can be proved that primitive matrices are the same as irreducible aperiodic non-negative matrices.

All statements of the Perron–Frobenius theorem for positive matrices remain true for primitive matrices. The same statements also hold for a non-negative irreducible matrix, except that it may possess several eigenvalues whose absolute value is equal to its spectral radius, so the statements need to be correspondingly modified. In fact the number of such eigenvalues is equal to the period.

Results for non-negative matrices were first obtained by Frobenius in 1912.