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.