Franklin's Notes

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:

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?




back to home page