Question: 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.
> 
> 
> 

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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.**

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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?

Answer: 
The Perron-Frobenius theorem is used in Google's Pagerank. It's one of the things that make sure that the algorithm works. [Here](http://www.math.pku.edu.cn/teachers/yaoy/Fall2011/lecture07.pdf) it's explained why the the theorem is useful, you have a lot of information with easy explanations on Google.
To check the stability of dynamical systems, one can try to search for a real-valued positive function named [Lyapunov function](https://en.wikipedia.org/wiki/Lyapunov_function).

If the dynamical system is [LTI](https://en.wikipedia.org/wiki/Linear_time-invariant_theory), 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](https://stanford.edu/class/ee363/lectures/pf.pdf).

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*](http://ieeexplore.ieee.org/abstract/document/1406483/)." IEEE Signal Processing Magazine 22.2 (2005): 62-75.