Login
This is a place holder for potential sponsor logos.
isomorphism of rings of real and complex matrices
Note that submatrix notation 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 $M_{2n \times 2n}(\mathbb{R})$ :
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
for $A,B \in M_{n \times n}(\mathbb{R})$ .
for $A,B \in M_{n \times n}(\mathbb{R})$ .
Let $A,B,C,D \in M_{n \times n}(\mathbb{R})$ such that
. Then $A+iB=C+iD$ . Therefore, $A=C$ and $B=D$ . Hence,
. 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
, it follows that $\varphi$ is surjective.
Let $A,B,C,D \in M_{n \times n}(\mathbb{R})$ . Then
and
It follows that $\varphi$ is an isomorphism. 
isomorphism of rings of real and complex matrices is owned by Warren Buck.
None.
[ View all 1 ]