definition of \(m \times n\) quaternion matrices topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of quaternions division associative algebra.
- The reader knows a definition of %ring name% matrices space.
- The reader knows a definition of \(m \times n\) complex matrices topological space.
- The reader admits the proposition that the quaternions division associative algebra is 'rings - homomorphisms' isomorphic to a subring of the \(2 \times 2\) complex matrices ring.
Target Context
- The reader will have a definition of \(m \times n\) quaternion 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:
\( m\): \(\in \mathbb{N} \setminus \{0\}\)
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\( \{M\}\): \(= \text{ the } \mathbb{H} \text{ matrices space }\)
\(*M_{m \times n} (\mathbb{H})\): \(= \{M \in \{M\} \vert M \in \text{ the } m \times n \text{ matrices } \}\), \(\in \{\text{ the topological spaces }\}\)
\( M_{2 m \times 2 n} (\mathbb{C})\): \(= \text{ the complex matrices topological space }\)
\( f\): \(: \mathbb{H} \to R\), \(= \text{ the 'rings - homomorphisms' isomorphism }\) defined in the proposition that the quaternions division associative algebra is 'rings - homomorphisms' isomorphic to a subring of the \(2 \times 2\) complex matrices ring
\( f'\): \(: M_{m \times n} (\mathbb{H}) \to M_{2 m \times 2 n} (\mathbb{C})\) which replaces each component of each element of \(M_{m \times n} (\mathbb{H})\) with the image of it under \(f\)
\( f'^`\): \(: M_{m \times n} (\mathbb{H}) \to f' (M_{m \times n} (\mathbb{H}))\), where \(f' (M_{m \times n} (\mathbb{H}))\) is the topological subspace of \(M_{2 m \times 2 n} (\mathbb{C})\), \(= \text{ the codomain restriction of } f'\)
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Conditions:
\(f'^` \in \{\text{ the homeomorphisms }\}\)
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When \(m = n\), \(M_{m \times n} (\mathbb{H})\) is often denoted as \(M_n (\mathbb{H})\).
2: Note
In other words, \(M_{m \times n} (\mathbb{H})\) is given the topology that makes \(f'^`\) a homeomorphism, which is valid, because \(f'^`\) is a bijection: any \(S \subseteq M_{m \times n} (\mathbb{H})\) is open if and only if \(f'^` (S) \subseteq f' (M_{m \times n} (\mathbb{H}))\) is open.
Let us see that \(f' (M_{m \times n} (\mathbb{H}))\) is a closed subspace of \(M_{2 m \times 2 n} (\mathbb{C})\).
\(f' (M_{m \times n} (\mathbb{H}))\) is nothing but the subset of \(M_{2 m \times 2 n} (\mathbb{C})\) such that each \(2 \times 2\) block of each element has the form, \(\begin{pmatrix} a + b i & c + d i \\ - c + d i & a - b i \end{pmatrix}\).
Let \(p \in M_{2 m \times 2 n} (\mathbb{C}) \setminus f' (M_{m \times n} (\mathbb{H}))\) be any.
A \(2 \times 2\) block of \(p\) has the form, \(\begin{pmatrix} a + b i & c + d i \\ - c' + d' i & a' - b' i \end{pmatrix}\) where \(a \neq a'\), \(b \neq b'\), \(c \neq c'\), or \(d \neq d'\).
Let us suppose that \(a \neq a'\).
There is an \(\epsilon \in \mathbb{R}\) such that \(0 \lt \epsilon\) and \(B_{a, \epsilon} \cap B_{a', \epsilon} = \emptyset\).
There is the homeomorphism, \(h: M_{2 m \times 2 n} (\mathbb{C}) \to \mathbb{R}^{2 (4 m n)}\), by which the topology of \(M_{2 m \times 2 n} (\mathbb{C})\) is defined.
\(h (p)\) has the form, \((..., a, b, - c', d', ..., c, d, a', - b', ...)\), where \(a\) is the \(j\) component and \(a'\) is the \(l\) component.
Let us take \(B_{h (p), \epsilon} \subseteq \mathbb{R}^{2 (4 m n)}\).
For each \(r \in B_{h (p), \epsilon}\), \(r^j \neq r^l\), because \(r^j \in B_{a, \epsilon}\) and \(r^l \in B_{a', \epsilon}\).
So, \(h^{- 1} (B_{h (p), \epsilon}) \subseteq M_{2 m \times 2 n} (\mathbb{C}) \setminus f' (M_{m \times n} (\mathbb{H}))\), because \(h^{- 1} (r)\) has the concerned \(2 \times 2\) block as \(\begin{pmatrix} r^j + r^{j + 1} i & r^{l - 2} + r^{l - 1} i \\ r^{j + 2} + r^{j + 3} i & r^l + r^{l + 1} i \end{pmatrix}\), where \(r^j \neq r^l\).
\(h^{- 1} (B_{h (p), \epsilon})\) is an open neighborhood of \(p\), because \(h\) is a homeomorphism.
So, \(M_{2 m \times 2 n} (\mathbb{C}) \setminus f' (M_{m \times n} (\mathbb{H}))\) is open, by the local criterion for openness.
Likewise, when \(b \neq b'\), \(c \neq c'\), or \(d \neq d'\).
So, \(f' (M_{m \times n} (\mathbb{H})) \subset M_{2 m \times 2 n} (\mathbb{C})\) is closed.
When \(m = n\), \(M_n (\mathbb{H})\) is a ring and \(f'^`\) is a 'rings - homomorphisms' isomorphism, by the proposition that the ring of the \(n \times n\) quaternion matrices is 'rings - homomorphisms' isomorphic to a subring of the \(2 n \times 2 n\) complex matrices, via the between-the-quaternions-division-associative-algebra-and-a-subring-of-\(2 \times 2\)-complex-matrices 'rings - homomorphisms' isomorphism, and the multiplication, \(m: M_n (\mathbb{H}) \times M_n (\mathbb{H}) \to M_n (\mathbb{H})\), is continuous, by the proposition that the multiplication on any multiplication-possible finite-product of quaternion matrices topological spaces is continuous.
Let us see that the quaternion-conjugation-transposition, \(c: M_n (\mathbb{H}) \to M_n (\mathbb{H}), M \mapsto M^*\), is continuous.
Let us see that \(c = {f'^`}^{- 1} \circ c' \circ f'^`\), where \(c'\) is the Hermitian-conjugation on \(f' (M_n (\mathbb{H})) \subseteq M_{2 n} (\mathbb{C})\).
Let \(M \in M_n (\mathbb{H})\) be any and \(M' := f'^` (M)\).
The \((j, l)\)-\(2 \times 2\)-block of \(M'\) is \(\begin{pmatrix} M'^{2 (j - 1) + 1}_{2 (l - 1) + 1} & M'^{2 (j - 1) + 1}_{2 (l - 1) + 2} \\ M'^{2 (j - 1) + 2}_{2 (l - 1) + 1} & M'^{2 (j - 1) + 2}_{2 (l - 1) + 2} \end{pmatrix} = \begin{pmatrix} a^j_l + b^j_l i & c^j_l + d^j_l i \\ - c^j_l + d^j_l i & a^j_l - b^j_l i \end{pmatrix}\), where \(a^j_l + b^j_l i + c^j_l j + d^j_l k\) is the \((j, l)\)-component of \(M\).
\(c' (M')^p_q = \overline{M'^q_p}\), so, the \((l, j)\)-\(2 \times 2\)-block of \(c' (M')\) is \(\begin{pmatrix} c' (M')^{2 (l - 1) + 1}_{2 (j - 1) + 1} & c' (M')^{2 (l - 1) + 1}_{2 (j - 1) + 2} \\ c' (M')^{2 (l - 1) + 2}_{2 (j - 1) + 1} & c' (M')^{2 (l - 1) + 2}_{2 (j - 1) + 2} \end{pmatrix} = \begin{pmatrix} \overline{M'^{2 (j - 1) + 1}_{2 (l - 1) + 1}} & \overline{M'^{2 (j - 1) + 2}_{2 (l - 1) + 1}} \\ \overline{M'^{2 (j - 1) + 1}_{2 (l - 1) + 2}} & \overline{M'^{2 (j - 1) + 2}_{2 (l - 1) + 2}} \end{pmatrix} = \begin{pmatrix} a^j_l - b^j_l i & - c^j_l - d^j_l i \\ c^j_l - d^j_l i & a^j_l + b^j_l i \end{pmatrix}\).
So, the \((l, j)\)-component of \({f'^`}^{- 1} \circ c' \circ f'^` (M)\) is \(a^j_l - b^j_l i - c^j_l j - d^j_l k = \overline{a^j_l + b^j_l i + c^j_l j + d^j_l k}\).
So, \(({f'^`}^{- 1} \circ c' \circ f'^` (M))^l_j = \overline{M^j_l}\), which means that \({f'^`}^{- 1} \circ c' \circ f'^` (M) = M^* = c (M)\).
\(f'^`\) is continuous, because \(f'^`\) is a homeomorphism.
\(c'\) is continuous, because \(c'\) is the restriction of the Hermitian-conjugation on \(M_{2 n} (\mathbb{C})\), which is continuous, by Note for the definition of \(m \times n\) complex matrices topological space, and the proposition that any restriction of any continuous map on the domain and the codomain is continuous applies.
\({f'^`}^{- 1}\) is continuous, because \(f'^`\) is a homeomorphism.
So, \(c = {f'^`}^{- 1} \circ c' \circ f'^`\) 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.
An expected question is why \(M_{m \times n} (\mathbb{H})\) is not made homeomorphic to \(\mathbb{R}^{4 m n}\) by \(f'^`: M \mapsto (a^1_1, b^1_1, c^1_1, d^1_1, ..., a^m_1, b^m_1, c^m_1, d^m_1, ..., a^1_n, b^1_n, c^1_n, d^1_n, ..., a^m_n, b^m_n, c^m_n, d^m_n)\), where \(M^j_l = a^j_l + b^j_l i + c^j_l j + d^j_l k\), which is of course possible, but is not so convenient, because it is not (at least directly) connected with convenient \(f\). In fact, when \(m = n\), \(f'^`\) is \(f''\) in the proposition that the ring of the \(n \times n\) quaternion matrices is 'rings - homomorphisms' isomorphic to a subring of the \(2 n \times 2 n\) complex matrices, via the between-the-quaternions-division-associative-algebra-and-a-subring-of-\(2 \times 2\)-complex-matrices 'rings - homomorphisms' isomorphism.
