Chapter 5: Eigenvalues and Eigenvectors

So far, we have treated matrices as arrays of numbers used in operations like multiplication. We now shift perspective to see matrices as operators that perform linear transformations. When a matrix multiplies a vector, it can stretch, shrink, or rotate it, mapping it to a new position in space.

This raises a question: for a given transformation, are there any vectors whose direction is unchanged? The answer is found in the study of eigenvalues and eigenvectors. An eigenvector is a vector that is only scaled by a transformation, not pointed in a new direction. The corresponding eigenvalue is the scalar factor of that scaling. This relationship is compactly expressed by the equation:

$Av = \lambda v$

Here, $A$ is the transformation matrix, $v$ is an eigenvector, and $\lambda$ (lambda) is its corresponding eigenvalue.

Throughout this chapter, we will cover the following topics:

• How to view matrices as functions that perform linear transformations.
• The formal definition of eigenvalues and eigenvectors.
• The geometric meaning behind this vector and scalar pairing.
• The method for calculating eigenvalues using the characteristic equation.
• How to use NumPy to compute eigenvalues and eigenvectors for a given matrix.