Applications of Eigen-analysis

Markov Chains & Dynamic Systems

We've developed the complete theory of eigenvalues and eigenvectors ( $Av=\lambda v$ ) and the powerful computational tool of diagonalization ( $A = PDP^{-1}$ ).

Now, we put it to work. The true power of eigen-analysis is in understanding systems that evolve over time. These are called dynamic systems. We will explore one of the most famous and useful examples: Markov Chains.

What is a Dynamic System?

A dynamic system is any system that changes from one state to another over discrete time steps according to a fixed rule. If that rule is a linear transformation, we can model it with a matrix.

Let $x_k$ be the state of a system at time step $k$ . The state at the next time step, $x_{k+1}$ , is given by:

x_{k+1} = A x_k

$x_0$ is our initial state.
$x_1 = A x_0$
$x_2 = A x_1 = A (A x_0) = A^2 x_0$
In general: $x_k = A^k x_0$

The state of the system at any future time $k$ is determined by the $k$ -th power of the transformation matrix $A$ applied to the initial state $x_0$ .

Application: Markov Chains

A Markov Chain is a specific type of dynamic system used to model probabilities. It describes the movement of something between a finite number of states.

The Rules of a Markov Matrix (or Transition Matrix):

All entries are between 0 and 1 (they represent probabilities).
The entries in each column sum to 1 (the total probability of moving from a given state must be 100%).

Example: A Simple Market Model

Imagine a city where people choose between two streaming services, "StreamFlix" and "ConnectPlus." Each month, some people switch.

StreamFlix keeps 90% of its customers, and 10% switch to ConnectPlus.
ConnectPlus keeps 80% of its customers, and 20% switch to StreamFlix.

We model this with a transition matrix $A$ :

A = \begin{bmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{bmatrix}

Let $x_0 = [0.6, 0.4]^T$ be the initial market share. After one month:

x_1 = A x_0 = \begin{bmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{bmatrix} \begin{bmatrix} 0.6 \\ 0.4 \end{bmatrix} = \begin{bmatrix} 0.62 \\ 0.38 \end{bmatrix}

StreamFlix will have 62% and ConnectPlus will have 38%.

The Big Question: What Happens in the Long Run?

The **steady state** of this system is the eigenvector corresponding to the eigenvalue

\lambda = 1

We need to find the eigenvector for $\lambda=1$ by solving $(A - I)v = 0$ .

A - I = \begin{bmatrix} 0.9-1 & 0.2 \\ 0.1 & 0.8-1 \end{bmatrix} = \begin{bmatrix} -0.1 & 0.2 \\ 0.1 & -0.2 \end{bmatrix}

This gives the equation $-0.1s + 0.2c = 0$ , which simplifies to $s = 2c$ . The eigenvector is of the form $[2c, c]$ . We can choose `c=1`, giving a basis vector $v_1 = [2, 1]$ .

To make this a probability vector (components sum to 1), we normalize it:

x_{ss} = \begin{bmatrix} 2/3 \\ 1/3 \end{bmatrix} \approx \begin{bmatrix} 0.667 \\ 0.333 \end{bmatrix}

Conclusion: The market will stabilize with StreamFlix holding 2/3 of the market and ConnectPlus holding 1/3, regardless of the initial state.

Summary: Eigen-analysis in Action

Dynamic systems $x_{k+1} = Ax_k$ can be understood by analyzing the matrix `A`.
Diagonalization ( $A^k = PD^kP^{-1}$ ) is key to computing long-term states.
Markov Chains model probabilistic state changes.
The steady state of a Markov chain is the eigenvector for the eigenvalue $\lambda = 1$ .
Other eigenvalues with magnitude less than 1 determine the rate of convergence to the steady state.

**Up Next:** We will explore the special case of **Symmetric Matrices** and the **Spectral Theorem**.