## The Eigenvalue Problem (A Refresher)

One of the most interesting functions in mathematical analysis and calculus is the exponential function, $\displaystyle x(t) = e^t$.

For one, this is the only non-trivial function that is it’s own derivative (the trivial function would be $x(t)=0$). That is, it is the solution to this very simple differential equation:

$\displaystyle \frac{dx}{dt} = x$ , with initial conditions $x(0)=1$

(Reminder: You solve it by separating the variables and then integrating. See here for a demonstration).

This “definition” of the exponential has interesting implications when you turn away from a simple equation and look toward systems of equations.

Consider this matrix equation:

$\mathbf{x'} = \mathbf{Ax}$

… where $x_i'(t)$ is in the $i^{th}$ place of the vector $\mathbf{x'}$, $x_i(t)$ is in the $i^{th}$ place of the vector $\mathbf{x}$, $i$ goes from 1 to n, and $\mathbf{A}$ is just any $n \times n$ matrix.

The form of this matrix equation is identical to the differential equation written above. Indeed, you might want to just say that it’s a generalization to arbitrary dimensions. It might also invite you to suppose, as was done previously, a solution involving exponentials. Except this time you must account for the vectorized nature of the equation. So try this:

$\displaystyle \mathbf{x} = e^{at}\mathbf{z}$, where $\mathbf{z}$ is unknown at this point.

Differentiating with respect to t, we get:

$\displaystyle \mathbf{x'} = ae^{at}\mathbf{z}$.

Plugging into our matrix equation:

$\displaystyle ae^{at}\mathbf{z} = e^{at}\mathbf{Az}$.

Since the exponentials can’t attain zero, we can divide them out, getting the following:

$\mathbf{Az}=a\mathbf{z}$

This particular equation is the so-called eigenvalue problem. Waxing ecstatic about the eigenvalue problem is all too easy, because it’s quite a profound little tool, both in mathematics and in science. It gives rise to the ideas of operator calculus, wherein one thinks of a complicated system of equations as an operator acting on a simpler system of equations. This allows you to reduce the system so that it’s (hopefully) solveable.

In the case of the eigenvalue problem, we’ve transformed a system of differential equations into a matrix operation that simply scales any vector it’s applied to by $a$. Solving this problem involves application of what universities sometimes call ‘linear algebra’. Solving it involves knowledge which I’ll slightly gloss over, but which you can look up easily in a linear algebra text or online.

Subtracting everything to one side, we get:

$\mathbf{Az}-a\mathbf{z}=\mathbf{0}$

Using the fact that $\mathbf{Ix}=\mathbf{x}$, for any vector $\mathbf{x}$, where $\mathbf{I}$ is the identity matrix, we can write this equation as:

$\mathbf{Az}-a\mathbf{Iz}=\mathbf{0}$

Removing the common factor (which requires the previous step; can’t subtract a vector from a matrix!), the equation resolves to:

$\left(\mathbf{A}-a\mathbf{I} \right) \mathbf{z}=\mathbf{0}$

Now, we wish this equation to have a non-trivial solution–that is, one where $\mathbf{z} \neq \mathbf{0}$.

There’s a particular theorem in linear algebra (sometimes called the Invertible Matrix Theorem) which will guide us here. First, a quick definition.

A matrix $\mathbf{B}$ is said to be invertible if there exists a matrix $\mathbf{C}$ such that $\mathbf{BC}=\mathbf{CB}=\mathbf{I}$. The matrix $\mathbf{C}$ is called the inverse of $\mathbf{B}$ and is written $\mathbf{B^{-1}}$.

The Invertible Matrix Theorem states that a matrix $\mathbf{B}$ is invertible if and only if the matrix equation $\mathbf{Bx}=\mathbf{0}$ has only the trivial solution $\mathbf{x}=\mathbf{0}$. Which means if we want a non-trivial solution out of our equation, we need to ensure that $\mathbf{A}-a\mathbf{I}$, which we’ll just call $\mathbf{B}$, is not invertible. To ensure this, let’s take a look at a hypothetical inverse for $\mathbf{B}$.

Assume, for a moment, that $\mathbf{B}$ is the following:

$\mathbf{B} = \left(\begin{array}{cc}b_1 & b_2\\ b_3 & b_4\end{array}\right)$

You may recall that if $\mathbf{B}$ has an inverse, it must be:

$\mathbf{B^{-1}} = \frac{1}{\det{\mathbf{B}}} \left(\begin{array}{cc}b_4 & -b_2\\ -b_3 & b_1\end{array}\right)$

This rule generalizes to arbitrary finite dimensions. Which means if $\det{\mathbf{B}} = 0$, then the matrix $\mathbf{B}$ is not invertible, because you’re dividing by zero. Horrah! (Here, $\det$ is the determinant, in case you have not seen it before).

So we get this equation:

$\det{\mathbf{B}} = \det{\left(\mathbf{A}-a\mathbf{I}\right)} = 0$

The left side of this equation turns out to be a polynomial in a. Solving for its zeroes gives you your different scale factors, called eigenvalues (these are your a‘s). What do you do with them? Plug ’em back into the equation and solve for your $\mathbf{z}$‘s!

Since you already know an expression for $\mathbf{x}$ in terms of $\mathbf{z}$, you can now solve for various values of $\mathbf{x}$.