definition of Frobenius matrix norm
Topics
About: matrix
The table of contents of this article
Starting Context
- The reader knows a definition of %ring name% matrices space.
- The reader knows a definition of norm on real or complex vectors space.
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: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( M (F, m \times n)\): \(= \text{ the } m \times n F \text{ matrices space }\), with the canonical vectors space structure
\(*\Vert \bullet \Vert_F\): \(: M (F, m \times n) \to \mathbb{R}, M \mapsto \sqrt{\sum_{j \in \{1, ..., m\}, l \in \{1, ..., n\}} \vert M^j_l \vert^2}\)
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Conditions:
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2: Note
It is obviously a norm, because it is the thing induced by the Euclidean norm or the complex Euclidean norm on the Euclidean vectors space or the complex Euclidean vectors space, \(F^{m n}\), to which \(M (F, m \times n)\) is 'vectors spaces - linear morphisms' isomorphic.
By the equivalence of norms for finite vectors space theorems (the proposition that any norms on any finite-dimensional real vectors space are equivalent and the proposition that any norms on any finite-dimensional complex vectors space are equivalent), 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.
