A definition of matrix norm induced by vector norms
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of normed vectors space.
- The reader knows a definition of matrix.
Target Context
- The reader will have a definition of matrix norm induced by vector norms.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Definition
For any normed real or complex column vectors spaces, \(V_1, V_2\), and any real or complex matrix, \(M\), as the representative of any linear map from \(V_1\) into \(V_2\), \(sup \{\Vert M v \Vert / \Vert v \Vert \vert v \in V_1, \Vert v \Vert \neq 0\}\), denoted as \(\Vert M \Vert\)
2: Note
As the definition depends on the norms of \(V_1, V_2\), it is meaningless to talk about the matrix norm in this sense without specifying the normed vectors spaces.
Although it is not clear what exactly the norm is for a matrix, for any finite-dimensional matrix, \(M\), by the equivalence of norms for finite vectors space theorem, \(r_1 \Vert M \Vert_F \le \Vert M \Vert \le r_2 \Vert M \Vert_F\) for some positive numbers, \(r_1, r_2 \in \mathbb{R}\), where \(\Vert \bullet \Vert_F\) denotes the Frobenius norm, and \(\Vert M v \Vert \le \Vert M \Vert \Vert v \Vert \le r_2 \Vert M \Vert_F \Vert v \Vert\), which is enough for many cases even with \(r_2\) unknown. Especially, \(\Vert M \Vert\) is always finite.
