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 %ring name% matrices space.
- The reader knows a definition of normed vectors space.
- The reader admits the proposition that for any map from any module with any \(d_1\)-elements basis into any module with any \(d_2\)-elements basis over any same ring, the map is linear if and only if the map is represented by the \(d_2 \times d_1\) ring matrix.
- The reader admits the proposition that any linear map from any finite-dimensional normed vectors space into any normed vectors space is continuous.
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The reader admits
the proposition that any linear map between any vectors metric spaces induced by any norms is continuous if and only if it is bounded .
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: 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
\( V_1\): \(\in \{\text{ the normed } n \text{ -dimensional } F \text{ column vectors spaces }\}\)
\( V_2\): \(\in \{\text{ the normed } m \text{ -dimensional } F \text{ column vectors spaces }\}\)
\(*\Vert \bullet \Vert\): \(: M (F, m \times n) \to \mathbb{R}, M \mapsto Sup (\{\Vert M v \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\})\)
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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.
\(\Vert \bullet \Vert\) is indeed into \(\mathbb{R}\), by the proposition that for any map from any module with any \(d_1\)-elements basis into any module with any \(d_2\)-elements basis over any same ring, the map is linear if and only if the map is represented by the \(d_2 \times d_1\) ring matrix, the proposition that any linear map from any finite-dimensional normed vectors space into any normed vectors space is continuous, and the proposition that any linear map between any vectors metric spaces induced by any norms is continuous if and only if it is bounded.
Let us see that \(\Vert \bullet \Vert\) is indeed a norm.
Let \(M_1, M_2 \in M (F, m \times n)\) be any.
1) \(0 \le \Vert M_1 \Vert\) with the equality holding if and only if \(M_1 = 0\): \(0 \le Sup (\{\Vert M_1 v \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\})\); if \(M_1 = 0\), \(Sup (\{\Vert M_1 v \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\}) = Sup (\{\Vert 0 \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\}) = Sup (\{0\}) = 0\); if \(Sup (\{\Vert M_1 v \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\}) = 0\), \(\Vert M_1 v \Vert = 0\) for each \(v\), so, \(M_1 v = 0\) for each \(v\), but by taking \(v = (0, ..., 0, 1, 0, ..., 0)^t\) where \(1\) is the \(j\)-th component, \(M_1 v\) is the \(j\)-th column of \(M_1\), so, the \(j\)-th column of \(M_1\) is \(0\), so, all the columns of \(M_1\) are \(0\), so, \(M_1 = 0\).
2) \(\Vert r M_1 \Vert = \vert r \vert \Vert M_1 \Vert\): \(\Vert r M_1 \Vert = Sup (\{\Vert r M_1 v \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\}) = Sup (\{\vert r \vert \Vert M_1 v \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\}) = \vert r \vert Sup (\{\Vert M_1 v \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\}) = \vert r \vert \Vert M_1 \Vert\).
3) \(\Vert M_1 + M_2 \Vert \le \Vert M_1 \Vert + \Vert M_2 \Vert\): \(\Vert M_1 + M_2\Vert = Sup (\{\Vert (M_1 + M_2) v \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\}) = Sup (\{\Vert M_1 v + M_2 v \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\}) \le Sup (\{(\Vert M_1 v \Vert + \Vert M_2 v \Vert) / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\}) = Sup (\{(\Vert M_1 v \Vert / \Vert v \Vert + \Vert M_2 v \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\}) \le Sup (\{\Vert M_1 v \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\})\}) + Sup (\{\Vert M_2 v \Vert / \Vert v \Vert \vert v \in V_1 \setminus \{0\}\})\), by the proposition that for any partially-ordered ring, any finite number of subsets with any same index set, and the subset as the sum of the subsets with the same index set, if the supremums of the subsets exist, the sum of the supremums of the subsets is an upper bound of the subset and if furthermore the supremum of the subset exists, the supremum of the subset is equal to or smaller than the sum of the supremums of the subsets and if the infimums of the subsets exist, the sum of the infimums of the subsets is a lower bound of the subset and if furthermore the infimum of the subset exists, the infimum of the subset is equal to or larger than the sum of the infimums of the subsets, \(= \Vert M_1 \Vert + \Vert M_2 \Vert\).
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 (the proposition that any norms on any finite-dimensional real vectors space are equivalent or the proposition that any norms on any finite-dimensional complex vectors space are equivalent), \(r_1 \Vert M \Vert_F \le \Vert M \Vert \le r_2 \Vert M \Vert_F\) for some positive numbers, \(r_1, r_2 \in \mathbb{R}\), where \(\Vert \bullet \Vert_F\) denotes the Frobenius matrix 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.
