Chapter 1: Vector Space and Inner Product. In the chapter mind map this icon labels Norm($\ell_1$, $\ell_2$, $\ell_\infty$, Std-$\ell_2$, Mahalanobis). The discussion below is excerpted and lightly edited from § Definition: Mahalanobis Norm (M-norm) in Mathematics for AI and Machine Learning.

For a vector $\mathbf{v} \in \mathbb{R}^d$ and a positive definite covariance matrix $\Sigma \in \mathbb{R}^{d \times d}$, the Mahalanobis norm (denoted as M-norm or Mahalanobis distance) is defined as

The M-norm accounts for correlations between dimensions through the covariance matrix $\Sigma$. When $\Sigma = I$ (identity matrix), it reduces to the standard Euclidean norm. The unit ball of the M-norm is a rotated and scaled ellipse (or ellipsoid) whose orientation and shape are determined by the eigenvectors and eigenvalues of $\Sigma$. In machine learning, the M-norm is used in anomaly detection, clustering (e.g., Gaussian mixture models), and when data has correlated features.