Matrix Norm - Equivalence of Norms

Equivalence of Norms

For any two vector norms ||·||_α and ||·||_β, we have

for some positive numbers r and s, for all matrices A in . In other words, all norms on are equivalent; they induce the same topology on . This is true because the vector space has the finite dimension .

Moreover, for every vector norm on, there exists a unique positive real number such that is a sub-multiplicative matrix norm for every .

A sub-multiplicative matrix norm ||·||_α is said to be minimal if there exists no other sub-multiplicative matrix norm ||·||_β satisfying ||·||_β ≤ ||·||_α.