Induced Norms

Abstract

In many problems we shall be concerned at the same time with norms of vectors and matrices. It would seem unwise if we used completely unrelated norms for the vectors and matrices. It turns out to be convenient to have a matrix norm “induced” by the vector norm. This means that we require a theorem:

Theorem 3.1.If n(x) is a vector norm satisfying the vector norm axioms then for any matrixA,

$$m_n \left( A \right) = m\left( A \right) = \sup \frac{{n\left( {Ax} \right)}}{{n\left( x \right)}},$$

where the supremum is over all non-zero vectorsx, satisfies the matrix norm axioms and is called the norm induced by n(x).