Building on our understanding of matrices as linear transformations, this chapter focuses on special vectors whose direction is invariant under the transformation, only undergoing scaling. These vectors are known as eigenvectors, and the scaling factor is the corresponding eigenvalue.

You will start by learning the formal definition captured by the equation $Ax = \lambda x$. We will then explore the geometric meaning behind this equation, visualizing how eigenvectors represent the axes of transformation. You'll learn the standard methods for calculation: finding eigenvalues by solving the characteristic equation $det(A - \lambda I) = 0$ and subsequently determining the corresponding eigenvectors.

We will cover eigen-decomposition, the process of expressing certain matrices in terms of their eigenvalues and eigenvectors, often represented as $A = PDP^{-1}$. A significant application in machine learning is Principal Component Analysis (PCA), which uses eigenvectors for dimensionality reduction; we will examine this connection. Finally, you will see how to perform these calculations efficiently using Python's NumPy library.