What is the implication of Perron Frobenius Theorem?

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?

The Perron-Frobenius theorem is used in Google's Pagerank. It's one of the things that make sure that the algorithm works. Here it's explained why the the theorem is useful, you have a lot of information with easy explanations on Google.

For example, there are important applications in the theory of finite Markov chains. Among other things, it implies that a Markov chain with strictly positive transition probabilities has a unique stationary distribution.

First of all, the conclusion is more interesting if you also try an example where the matrix entries are still real, but not positive. For example, if $A$ is the $2\times2$ rotation matrix $$A=\pmatrix{\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta}$$ then you can check that the eigenvectors are no longer real, and there's no longer a unique eigenvalue of maximum modulus: the two eigenvalues are $e^{i\theta}$ and $e^{-i\theta}$.

I second Robert Israel's nomination of Markov chains as a great application. I'll add the following, though. (I'll remark that you don't actually need the matrix to have positive entries - just for some power of it to have positive entries.) Suppose you have a finite connected graph (or strongly connected digraph) such that the gcd of the cycle lengths is $1$. Then if $A$ is the adjacency matrix for the graph, some power $A^r$ will have positive entries so Perron-Frobenius applies. Since the entries of $A^n$ count paths in the graph, we conclude that for any pair of vertices in the graph, the number of paths of length $n$ between them is $c\lambda_1^n(1+o(1))$, where $\lambda_1$ is the Perron-Frobenius eigenvalue and $c$ is a computable constant. Applications of this particular consequence of Perron-Frobenius include asymptotic growth rate results for "regular languages," which are defined in terms of paths on graphs.

In particular, there is a cute formula giving the $n$-th Fibonacci number as $$F_n=\left\langle \frac1{\sqrt5}\phi^n\right\rangle$$ where $\phi$ is the "golden ratio" $(1+\sqrt5)/2$ and $\langle\cdot\rangle$ denotes "closest integer." This formula, and many others like it, become less surprising once you check that $$\pmatrix{1&1\\1&0}^n=\pmatrix{F_{n+1}&F_n\\ F_n&F_{n-1}}$$ and note that $\phi$ is the Perron-Frobenius eigenvalue (and the other eigenvalue is smaller than $1$ in absolute value.)

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.

One of the nice things about it is that for $v$ a vector with all positive entries, if we let $v_k = A^k v$, then $\lim_{k\to\infty} \frac{v^k}{|| v_k ||}$ exists and is an eigen vector for the Perron-Frobenius eigen value.