definition of \(m \times n\) complex matrices topological space
Topics
About: topological 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 complex Euclidean topological space.
Target Context
- The reader will have a definition of \(m \times n\) complex matrices topological space.
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:
\( m\): \(\in \mathbb{N} \setminus \{0\}\)
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\( \{M\}\): \(= \text{ the } \mathbb{C} \text{ matrices space }\)
\(*M_{m \times n} (\mathbb{C})\): \(= \{M \in \{M\} \vert M \in \text{ the } m \times n \text{ matrices } \}\), \(\in \{\text{ the topological spaces }\}\)
\( \mathbb{C}^{m n}\): \(= \text{ the complex Euclidean topological space }\)
\( f\): \(: M_{m \times n} (\mathbb{C}) \to \mathbb{C}^{m n}, M \mapsto (M^1_1, ..., M^m_1, ..., M^1_n, ..., M^m_n)\)
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Conditions:
\(f \in \{\text{ the homeomorphisms }\}\)
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When \(m = n\), \(M_{m \times n} (\mathbb{C})\) is often denoted as \(M_n (\mathbb{C})\).
2: Note
In other words, \(M_{m \times n} (\mathbb{C})\) is given the topology that makes \(f\) a homeomorphism, which is valid, because \(f\) is a bijection: any \(S \subseteq M_{m \times n} (\mathbb{C})\) is open if and only if \(f (S) \subseteq \mathbb{C}^{m n}\) is open.
When \(m = n\), \(M_n (\mathbb{C})\) is a ring, by the definition of ring of \(n \times n\) ring matrices, and the multiplication, \(m: M_n (\mathbb{C}) \times M_n (\mathbb{C}) \to M_n (\mathbb{C})\), is continuous, by the proposition that the multiplication on any multiplication-possible finite-product of complex matrices topological spaces is continuous.
Let us see that the Hermitian-conjugation, \(c: M_n (\mathbb{C}) \to M_n (\mathbb{C}), M \mapsto M^*\), is continuous.
There is the homeomorphism, \(h: M_n (\mathbb{C}) \to \mathbb{C}^{n^2} \to \mathbb{R}^{2 n^2}\), by which the topology of \(M_n (\mathbb{C})\) is defined.
\(c = h^{- 1} \circ h \circ c \circ h^{- 1} \circ h\).
Let us see that \(h \circ c \circ h^{- 1}: \mathbb{R}^{2 n^2} \to \mathbb{R}^{2 n^2}\) is continuous.
\(\mathbb{R}^{2 n^2}\) is a topological subspace of the \(C^{\infty}\) manifold, \(\mathbb{R}^{2 n^2}\), and by the standard chart for \(\mathbb{R}^{2 n^2}\), each component of each \(v \in \mathbb{R}^{2 n^2}\) becomes the real or imaginary part of a component of \(h^{- 1} (v)\), each component of \(c (h^{- 1} (v))\) is the complex-conjugate of a component of \(h^{- 1} (v)\), whose real and imaginary parts are a component of \(v\) and the minus of a component of \(v\), and those real and imaginary parts of the components of \(c (h^{- 1} (v))\) become the components of \(h \circ c \circ h^{- 1} (v)\), so, the components of the components function of \(h \circ c \circ h^{- 1}\) are some components and the minuses of some components, which are continuous.
So, \(h \circ c \circ h^{- 1}\) is continuous, by the proposition that any topological spaces map is continuous at any point if the spaces are the subspaces of some \(C^\infty\) manifolds and there are some charts of the manifolds around the point and the point image and a map between the chart open subsets which (the map) is restricted to the original map whose (the chart open subsets map's) coordinates function's restriction is continuous.
\(h\) is continuous, because \(h\) is a homeomorphism.
\(h^{- 1}\) is continuous, because \(h\) is a homeomorphism.
So, \(c = h^{- 1} \circ h \circ c \circ h^{- 1} \circ h\) is continuous, by the proposition that for any maps between any arbitrary subspaces of any topological spaces continuous at any corresponding points, the composition is continuous at the point.
