TITLE: What is the implication of Perron Frobenius Theorem?
QUESTION [22 upvotes]: The Perron Frobenius theorem states:

Any square matrix $A$ with positive entries has a unique eigenvector with
  positive entries (up to a multiplication by a positive scalar), and
  the corresponding eigenvalue has multiplicity one and is strictly
  greater than the absolute value of any other eigenvalue.


So I tempted fate using this matrix:
$$ A =\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$$
I find my eigenvalues to be $\lambda_1 = 0, \lambda_2 = 2$
Now I find my eigenvector, taking $v_{11} = 1$, $v_{21} = 1$
I find 
$v_1 = \begin{bmatrix} v_{11} & v_{12} \end{bmatrix}$ = $\begin{bmatrix} 1 & -1 \end{bmatrix}$
$v_1 = \begin{bmatrix} v_{21} & v_{22} \end{bmatrix}$ = $\begin{bmatrix} 1 & 1 \end{bmatrix}$
This verifies the Perron-Frobenius theorem.

Now what is the great implication that every positive square matrix has a real eigenvector with an eigenvalue that is the largest of all eigenvalues?
Can someone show me an application of this theorem?

REPLY [2 votes]: To check the stability of dynamical systems, one can try to search for a real-valued positive function named Lyapunov function.
If the dynamical system is LTI, that is, of the format 
$$ \dot{x}(t) = Ax(t) $$ 
or 
$$ x(k+1) = Ax(k),$$
one can restrict the search for quadratic Lyapunov functions, that is, for functions $V: \mathbb{R}^n \rightarrow \mathbb{R}$ of the format $V(x) = x^T P x$. 
If the LTI system is also positive (that is, the state is always guaranteed to be in the positive orthant of $\mathbb{R}^n$), it is possible to use Perron-Frobenius theorem to prove that this search can be restricted to linear functions, simplifying even more the problem of stability analysis. For details, see slides 31-37 of Prof. Boyd's lecture notes on Perron-Frobenius theory.
You can also check the following reference for an embracing survey of applications:
Pillai, S. Unnikrishna, Torsten Suel, and Seunghun Cha. "The Perron-Frobenius theorem: some of its applications." IEEE Signal Processing Magazine 22.2 (2005): 62-75.