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: 
I would like to add an engineering application. I am a researcher from a wireless communication background. One of the most fundamental problem in our area would be the so-called Power Control wherein a mobile tower transmitting individual data to the several mobiles has to minimize its own power consumption meanwhile ensuring a minimum level of signal quality at the mobile. This can be formalized as a non-convex optimization problem. However, using some theoretical tools based on the perron-frobenius theory, you can find a simple, iterative algorithm which finds the true solution to this problem. This result has been a break through in our field. The perron-frobenius eigenvector will be the sort of numbers (positive) which states how much power the mobile tower has to use to serve its users. A well-cited paper in this regard is "A framework for uplink power control in cellular radio systems", in case you are interested to read more about this.
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.