Computing Eigenvectors

In Chapter 4, we've established the groundwork by introducing eigenvectors and eigenvalues, essential tools in the linear algebra toolkit for machine learning. Now, let's delve into the mechanics of computing eigenvectors, a crucial step that allows us to unlock the insights they provide.

At its core, the process of finding eigenvectors involves solving the equation $A\mathbf{v} = \lambda\mathbf{v}$, where $A$ is a matrix, $\lambda$ is an eigenvalue, and $\mathbf{v}$ is the corresponding eigenvector. This equation tells us that when matrix $A$ operates on vector $\mathbf{v}$, the output is simply a scaled version of $\mathbf{v}$ itself. This property is what makes eigenvectors so valuable in simplifying complex systems, like those encountered in machine learning.

To compute eigenvectors, we first need to determine the eigenvalues of matrix $A$. This typically involves finding the roots of the characteristic polynomial, derived from the determinant equation $\det(A - \lambda I) = 0$, where $I$ is the identity matrix of the same dimension as $A$. Solving this polynomial yields the eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$.

Once we have the eigenvalues, the next step is to find the eigenvectors for each eigenvalue. For a given eigenvalue $\lambda_i$, we substitute it back into the equation $(A - \lambda_i I)\mathbf{v} = 0$. This is a homogeneous system of linear equations, which can be solved using various methods such as Gaussian elimination or matrix row reduction.

Let's go through a practical example to solidify this process. Suppose we have a 2x2 matrix:

$$A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$$

First, we calculate the characteristic polynomial:

$$\det(A - \lambda I) = \det\begin{pmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{pmatrix} = (4 - \lambda)(3 - \lambda) - 2 \cdot 1$$

Simplifying, we get:

$$(4 - \lambda)(3 - \lambda) - 2 = \lambda^2 - 7\lambda + 10 = 0$$

Solving this quadratic equation, we find the eigenvalues $\lambda_1 = 5$ and $\lambda_2 = 2$.

To find the eigenvectors, we start with $\lambda_1 = 5$:

$$(A - 5I)\mathbf{v} = \begin{pmatrix} 4 - 5 & 1 \\ 2 & 3 - 5 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -1 & 1 \\ 2 & -2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

This simplifies to the system of equations:

$$-x + y = 0 \quad \text{(Equation 1)}$$

From Equation 1, we see that $x = y$. So, any vector of the form $\mathbf{v}_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}k$, where $k$ is a scalar, is an eigenvector corresponding to $\lambda_1 = 5$.

For $\lambda_2 = 2$, we have:

$$(A - 2I)\mathbf{v} = \begin{pmatrix} 4 - 2 & 1 \\ 2 & 3 - 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

This simplifies to:

$$2x + y = 0 \quad \text{(Equation 2)}$$

From Equation 2, $y = -2x$, so any vector of the form $\mathbf{v}_2 = \begin{pmatrix} 1 \\ -2 \end{pmatrix}k$, where $k$ is a scalar, is an eigenvector corresponding to $\lambda_2 = 2$.

By following these steps, we can efficiently compute eigenvectors for any square matrix. Understanding how to derive and interpret these vectors is crucial for applying techniques like Principal Component Analysis (PCA), which we will explore further in subsequent sections. In machine learning, eigenvectors help us reduce dimensionality, identify patterns, and optimize algorithms, making them indispensable for analyzing high-dimensional datasets.