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.
Read more about this topic: Matrix Norm
Famous quotes containing the word norms:
“There is a totalitarian regime inside every one of us. We are ruled by a ruthless politburo which sets ours norms and drives us from one five-year plan to another. The autonomous individual who has to justify his existence by his own efforts is in eternal bondage to himself.”
—Eric Hoffer (1902–1983)