Induced Norm

Theorem: Induced Norm Theorem

A norm $\Vert .\Vert \text{ on }{\mathbb{R}}^n$ that is induced by an inner product, there exists a symmetric positive definite (PD) matrix A such that:

$$\Vert x{\Vert }^2=x^{T}Ax,\forall x\in {\mathbb{R}}^n$$

• If A = I, the norm is Euclidean Norm
• If A is a diagonal matrix with positive entries, we get a weighted norm: $\Vert x{\Vert }_A=x^{T}Ax={\sum }_{i=1}^N{a}_i{x}_i^2$, where $a_i$ are diagonal elements of A (weights).
• If A is a covariance matrix, this norm corresponds to the Mahalanobis norm, common in statistics and machine learning.

Ellipsoid Norm Theorem

Every quadratic form defines an ellipsoid, showing a geometric interpretation of norms.