isomorphism of rings of real and complex matrices

Note that submatrix notation (http://planetmath.org/Submatrix) will be used within this entry. Also, for any positive integer $n$ , $M_{n\times n}(R)$ will be used to denote the ring of $n\times n$ matrices with entries from the ring $R$ , and $R_{n}$ will be used to denote the following subring of $M_{2n\times 2n}(\mathbb{R})$ :

$R_{n}=\left\{P\in M_{2n\times 2n}(\mathbb{R}):P=\left(\begin{array}[]{cc}A&B\\ -B&A\end{array}\right)\text{ for some }A,B\in M_{n\times n}(\mathbb{R})\right\}$

Theorem.

For any positive integer $n$ , $R_{n}\cong M_{n\times n}(\mathbb{C})$ .

Proof.

Define $\varphi\colon R_{n}\to M_{n\times n}(\mathbb{C})$ by $\displaystyle\varphi\left(\left(\begin{array}[]{cc}A&B\\ -B&A\end{array}\right)\right)=A+iB$ for $A,B\in M_{n\times n}(\mathbb{R})$ .

Let $A,B,C,D\in M_{n\times n}(\mathbb{R})$ such that $\displaystyle\varphi\left(\left(\begin{array}[]{cc}A&B\\ -B&A\end{array}\right)\right)=\varphi\left(\left(\begin{array}[]{cc}C&D\\ -D&C\end{array}\right)\right)$ . Then $A+iB=C+iD$ . Therefore, $A=C$ and $B=D$ . Hence, $\displaystyle\left(\begin{array}[]{cc}A&B\\ -B&A\end{array}\right)=\left(\begin{array}[]{cc}C&D\\ -D&C\end{array}\right)$ . It follows that $\varphi$ is injective.

Let $Z\in M_{n\times n}(\mathbb{C})$ . Then there exist $X,Y\in M_{n\times n}(\mathbb{R})$ such that $X+iY=Z$ . Since $\varphi\left(\left(\begin{array}[]{cc}X&Y\\ -Y&X\end{array}\right)\right)=X+iY=Z$ , it follows that $\varphi$ is surjective.

Let $A,B,C,D\in M_{n\times n}(\mathbb{R})$ . Then

$\begin{array}[]{rl}\displaystyle\varphi\left(\left(\begin{array}[]{cc}A&B\\ -B&A\end{array}\right)+\left(\begin{array}[]{cc}C&D\\ -D&C\end{array}\right)\right)&\displaystyle=\varphi\left(\left(\begin{array}[]% {cc}A+C&B+D\\ -B-D&A+C\end{array}\right)\right)\\ \\ &=A+C+i(B+D)\\ \\ &=A+iB+C+iD\\ \\ &\displaystyle=\varphi\left(\left(\begin{array}[]{cc}A&B\\ -B&A\end{array}\right)\right)+\varphi\left(\left(\begin{array}[]{cc}C&D\\ -D&C\end{array}\right)\right)\end{array}$

and

$\begin{array}[]{rl}\displaystyle\varphi\left(\left(\begin{array}[]{cc}A&B\\ -B&A\end{array}\right)\left(\begin{array}[]{cc}C&D\\ -D&C\end{array}\right)\right)&\displaystyle=\varphi\left(\left(\begin{array}[]% {cc}AC-BD&AD+BC\\ -AD-BC&AC-BD\end{array}\right)\right)\\ \\ &=AC-BD+i(AD+BC)\\ \\ &=(A+iB)(C+iD)\\ \\ &\displaystyle=\varphi\left(\left(\begin{array}[]{cc}A&B\\ -B&A\end{array}\right)\right)\varphi\left(\left(\begin{array}[]{cc}C&D\\ -D&C\end{array}\right)\right).\end{array}$

It follows that $\varphi$ is an isomorphism (http://planetmath.org/RingIsomorphism). ∎

Title	isomorphism of rings of real and complex matrices
Canonical name	IsomorphismOfRingsOfRealAndComplexMatrices
Date of creation	2013-03-22 16:17:15
Last modified on	2013-03-22 16:17:15
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	10
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 15-01
Classification	msc 15A33
Classification	msc 15A21