Vector norm

A vector norm is a function $\lVert\cdot\rVert: \mathbb{F}^n\to [0,\infty)$ that formalizes the notion of "length" in a vector space. It is defined by the following $3$ properties:

1. $\lVert x\rVert \geq 0$, and $\lVert x\rVert = 0$ iff $x=0$ (positive definiteness)
2. $\lVert x+y\rVert\leq \lVert x\rVert + \lVert y\rVert$ (triangle inequality)
3. $\lVert \alpha x\rVert = |\alpha|\lVert x\rVert$ for scalars $\alpha$ (homogeneity)

Here is a list of commonly used norms:

• The Euclidean norm, denoted $\lVert \cdot\rVert_2$ but often written as $\lVert \cdot\rVert$, is defined as $$\lVert x\rVert = \sqrt{x^\ast x} = \sqrt{\sum_{k=1}^n x_k^2}$$ and this is the most traditional notion of length. In $\mathbb{R}^n$, the unit ball under this norm is a sphere.

• The 1-norm, denoted $\lVert \cdot \rVert_1$, is defined as $$\lVert x\rVert_1 = \sum_{k=1}^n |x_k|$$ and the unit ball of the 1-norm in $\mathbb{R}^n$ is an n-dimensional polytope (e.g. a diamond in $\mathbb{R}^2$, and an octahedron in $\mathbb{R}^3$).

• The infinity-norm or sup norm, denoted $\lVert \cdot\rVert_\infty$, is defined as the maximum magnitude of the entries of $x$: $$\lVert x\rVert_\infty = \sup_{1\leq k \leq n} |x_k|$$ and the unit ball of the sup norm in $\mathbb{R^n}$ is an $n$-dimensional hypercube.

• The p-norms, denoted $\lVert \cdot\rVert_p$ for any value of $p\in (1,\infty)$, are defined as $$\lVert x\rVert_p = \bigg(\sum_{k=1}^n x_k^p\bigg)^{1/p}$$ The 1-norm and the infinity-norm are sometimes considered to be p-norms with $p=1$ and $p=\infty$, and the Euclidean norm is a p-norm with $p=2$.

Question. In $\mathbb R^n$, the unitary matrices are precisely the matrices which preserve distance under the 2-norm $\lVert \cdot\rVert_2$. What are the sets of matrices that preserve distance under each of the other norms listed above?

linear-algebra

vector-space

norm

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