A definition of Frobenius matrix norm
Topics
About: matrix
The table of contents of this article
Starting Context
- The reader knows a definition of matrix.
Target Context
- The reader will have a definition of Frobenius matrix norm.
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 real or complex matrix, \(M\), \(\sqrt{\sum_{j, k} \vert M_{j, k} \vert^2}\), denoted as \(\Vert M \Vert_F\)
2: Note
By the equivalence of norms for finite vectors space theorem, for any finite-dimensional matrix norm, \(\Vert \bullet \Vert\), \(r_1 \Vert M \Vert_F \le \Vert M \Vert \le r_2 \Vert M \Vert_F\) for some positive numbers, \(r_1, r_2 \in \mathbb{R}\), which can be used to evaluate \(\Vert M \Vert\), while what exactly \(r_1, r_2\) are does not matter in many cases.
