definition of \(n \times n\) symplectic matrices group
Topics
About: group
About: matrices 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 group.
Target Context
- The reader will have a definition of \(n \times n\) symplectic matrices group.
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:
\( \mathbb{H}\): \(= \text{ the quaternions division associative algebra }\)
\( \{M\}\): \(= \text{ the } \mathbb{H} \text{ matrices space }\)
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\(*Sp (n)\): \(= \{M \in \{M\} \vert M \in \{\text{ the } n \times n \text{ matrices such that } M^* = M^{- 1} \}\}\), where \({M^*}^j_l = \overline{M^l_j}\)
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Conditions:
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2: Note
Let us see that \(Sp (n)\) is indeed a group.
Let \(M_1, M_2, M_3 \in Sp (n)\) be any.
\((M_1 M_2)^* = {M_2}^* {M_1}^*\), because \({(M_1 M_2)^*}^j_l = \overline{(M_1 M_2)^l_j} = \overline{{M_1}^l_m {M_2}^m_j} = \overline{{M_2}^m_j} \overline{{M_1}^l_m}\), which is described in the definition of quaternions division associative algebra, \(= {{M_2}^*}^j_m {{M_1}^*}^m_l = ({M_2}^* {M_1}^*)^j_l\).
\({{M_1}^*}^* = M_1\), because \({{{M_1}^*}^*}^j_l = \overline{{{M_1}^*}^l_j} = \overline{\overline{{M_1}^j_l}} = {M_1}^j_l\).
\(M_1 M_2 \in Sp (n)\), because \((M_1 M_2)^* = {M_2}^* {M_1}^* = {M_2}^{- 1} {M_1}^{- 1}\), so, \((M_1 M_2)^* M_1 M_2 = {M_2}^{- 1} {M_1}^{- 1} M_1 M_2 = {M_2}^{- 1} I M_2 = {M_2}^{- 1} M_2 = I\) and \(M_1 M_2 (M_1 M_2)^* = M_1 M_2 {M_2}^{- 1} {M_1}^{- 1} = M_1 I {M_1}^{- 1} = M_1 {M_1}^{- 1} = I\), so, \((M_1 M_2)^* = (M_1 M_2)^{- 1}\).
1) \((M_1 \bullet M_2) \bullet M_3 = M_1 \bullet (M_2 \bullet M_3)\): by the proposition that for any ring, the multiplications of any matrices over the ring are associative.
2) \(i \in Sp (n)\) (called 'identity element') such that \(i \bullet M_1 = M_1 \bullet i = M_1\): the identity matrix, \(I\), is in \(Sp (n)\), because \(I^* = I = I^{-1}\), and \(I M_1 = M_1 I = M_1\).
3) \({M_1}^{- 1} \in Sp (n)\) (called 'inverse element of \(M_1\)') such that \({M_1}^{- 1} \bullet M_1 = M_1 \bullet {M_1}^{- 1} = I\): \({M_1}^* = {M_1}^{- 1} \in Sp (n)\), because \({{M_1}^*}^* = M_1\) and \({M_1}^* M_1 = I = M_1 {M_1}^*\), so, \({{M_1}^*}^{- 1} = M_1 = {{M_1}^*}^*\).
So, \(Sp (n)\) is a group.
In fact, only \(1\) of \(M M^* = I\) and \(M^* M = I\) implies \(M^* = M^{- 1}\), because while there is the 'rings - homomorphisms' isomorphism defined 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, \(f'': M_n (\mathbb{H}) \to f' (M_n (\mathbb{H}))\), supposing that \(M M^* = I\), \(f'' (M M^*) = f'' (I) = I\), but the left hand side is \(f'' (M) f'' (M^*)\), which implies that \(f'' (M^*) f'' (M) = I\), by the proposition that for any ring of \(n \times n\) field matrices, if any element multiplied by any another element from left or right is \(1\), the another element is the inverse of the element, so, \(f'' (M^* M) = f'' (M^*) f'' (M) = I\), which implies that \(M^* M = I\); likewise for supposing that \(M^* M = I\).
