description/proof of that 'rings - homomorphisms' isomorphism from ring of \(n \times n\) quaternion matrices onto subring of \(2 n \times 2 n\) complex matrices maps symplectic matrix to unitary matrix
Topics
About: ring
The table of contents of this article
Starting Context
- The reader knows a definition of ring of \(n \times n\) ring matrices.
- The reader knows a definition of \(n \times n\) symplectic matrices group.
- The reader knows a definition of unitary matrix.
- 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.
- The reader admits 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.
- The reader admits the proposition that the multiplication of any matrix made of any same size blocks by any matrix made of blocks of any multiplicable (with blocks of the former matrix) same size is blocks-wise.
- The reader admits the proposition that the canonical 'rings - homomorphisms' isomorphism from the quaternions division associative algebra onto the subring of the \(2 \times 2\) complex matrices ring maps the quaternion conjugate of any element to the Hermitian conjugate of the image of the element.
- The reader admits 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.
Target Context
- The reader will have a description and a proof of the proposition that the canonical 'rings - homomorphisms' isomorphism from the ring of the \(n \times n\) quaternion matrices onto the subring of the \(2 n \times 2 n\) complex matrices maps each symplectic matrix to a unitary matrix.
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:
\(\mathbb{H}\): \(= \text{ the quaternions division associative algebra }\)
\(R\): \(= \{\begin{pmatrix} c_1 & c_2 \\ \overline{- c_2} & \overline{c_1} \end{pmatrix} \vert c_1, c_2 \in \mathbb{C}\}\) with the canonical addition and multiplication
\(f\): \(: \mathbb{H} \to R\), \(= \text{ the 'rings - homomorphisms' isomorphism }\) 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
\(n\): \(\in \mathbb{N} \setminus \{0\}\)
\(M_n (\mathbb{H})\): \(= \text{ the ring of the } n \times n \text{ quaternions matrices }\) with the canonical addition and multiplication
\(M_{2 n} (\mathbb{C})\): \(= \text{ the ring of the } 2 n \times 2 n \text{ complex matrices }\) with the canonical addition and multiplication
\(f''\): \(: M_n (\mathbb{H}) \to f' (M_n (\mathbb{H}))\), \(= \text{ the 'rings - homomorphisms' isomorphism }\) 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
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Statements:
\(\forall M \in M_n (\mathbb{H}) \cap \{\text{ the symplectic matrices }\} (f'' (M) \in \{\text{ the unitary matrices }\})\)
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2: Proof
Whole Strategy: Step 1: see that the \((j, l)\)-\(2 \times 2\) block of \(f'' (M)^* f'' (M)\) is \(\sum_{m \in \{1, ..., n\}} f (M^m_j)^* f (M^m_l)\); Step 2: see that \(\sum_{m \in \{1, ..., n\}} f (M^m_j)^* f (M^m_l) = \delta^j_l I\); Step 3: conclude the proposition.
Step 1:
\({f'' (M)^*}^p_q = \overline{f'' (M)^q_p}\).
For any \(2 n \times 2 n\) matrix, \(N\), let \(N^{(j)}_{(l)}\) denote the \((j, l)\)-\(2 \times 2\) block of \(N\): \(N^{(j)}_{(l)} = \begin{pmatrix} N^{2 (j - 1) + 1}_{2 (l - 1) + 1} & N^{2 (j - 1) + 1}_{2 (l - 1) + 2} \\ N^{2 (j - 1) + 2}_{2 (l - 1) + 1} & N^{2 (j - 1) + 2}_{2 (l - 1) + 2} \end{pmatrix}\).
\((f'' (M)^* f'' (M))^{(j)}_{(l)} = {f'' (M)^*}^{(j)}_{(m)} {f'' (M)}^{(m)}_{(l)}\), by the proposition that the multiplication of any matrix made of any same size blocks by any matrix made of blocks of any multiplicable (with blocks of the former matrix) same size is blocks-wise.
\({f'' (M)^*}^{(j)}_{(m)} = \begin{pmatrix} {f'' (M)^*}^{2 (j - 1) + 1}_{2 (m - 1) + 1} & {f'' (M)^*}^{2 (j - 1) + 1}_{2 (m - 1) + 2} \\ {f'' (M)^*}^{2 (j - 1) + 2}_{2 (m - 1) + 1} & {f'' (M)^*}^{2 (j - 1) + 2}_{2 (m - 1) + 2} \end{pmatrix} = \begin{pmatrix} \overline{f'' (M)^{2 (m - 1) + 1}_{2 (j - 1) + 1}} & \overline{f'' (M)^{2 (m - 1) + 2}_{2 (j - 1) + 1}} \\ \overline{f'' (M)^{2 (m - 1) + 1}_{2 (j - 1) + 2}} & \overline{f'' (M)^{2 (m - 1) + 2}_{2 (j - 1) + 2}} \end{pmatrix}\).
On the other hand, \(f (M^m_j)^* = {f'' (M)^{(m)}_{(j)}}^* = \begin{pmatrix} f'' (M)^{2 (m - 1) + 1}_{2 (j - 1) + 1} & f'' (M)^{2 (m - 1) + 1}_{2 (j - 1) + 2} \\ f'' (M)^{2 (m - 1) + 2}_{2 (j - 1) + 1} & f'' (M)^{2 (m - 1) + 2}_{2 (j - 1) + 2} \end{pmatrix}^* = \begin{pmatrix} \overline{f'' (M)^{2 (m - 1) + 1}_{2 (j - 1) + 1}} & \overline{f'' (M)^{2 (m - 1) + 2}_{2 (j - 1) + 1}} \\ \overline{f'' (M)^{2 (m - 1) + 1}_{2 (j - 1) + 2}} & \overline{f'' (M)^{2 (m - 1) + 2}_{2 (j - 1) + 2}} \end{pmatrix}\).
So, \({f'' (M)^*}^{(j)}_{(m)} = f (M^m_j)^*\).
So, \((f'' (M)^* f'' (M))^{(j)}_{(l)} = {f'' (M)^*}^{(j)}_{(m)} {f'' (M)}^{(m)}_{(l)} = \sum_{m \in \{1, ..., n\}} f (M^m_j)^* f (M^m_l)\).
Step 2:
\(\sum_{m \in \{1, ..., n\}} f (M^m_j)^* f (M^m_l) = \sum_{m \in \{1, ..., n\}} f (\overline{M^m_j}) f (M^m_l)\), by the proposition that the canonical 'rings - homomorphisms' isomorphism from the quaternions division associative algebra onto the subring of the \(2 \times 2\) complex matrices ring maps the quaternion conjugate of any element to the Hermitian conjugate of the image of the element, \(= \sum_{m \in \{1, ..., n\}} f (\overline{M^m_j} M^m_l)\), because \(f\) is a 'rings - homomorphisms' isomorphism, \(= \sum_{m \in \{1, ..., n\}} f ({M^*}^j_m M^m_l)= f (\sum_{m \in \{1, ..., n\}} {M^*}^j_m M^m_l)\), because \(f\) is a 'rings - homomorphisms' isomorphism, \(= f (I^j_l)\), because \(M\) is symplectic, \(= f (\delta^j_l) = \delta^j_l I\).
Step 3:
So, \((f'' (M)^* f'' (M))^{(j)}_{(l)} = \delta^j_l I\), which means that \(f'' (M)^* f'' (M) = I\).
\(f'' (M)^* = f'' (M)^{- 1}\), 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, \(f'' (M)\) is unitary.
