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 $M_{2n\times 2n}(ℝ)$:

Theorem.

For any positive integer $n$, $R_n\mathrm{\cong }{M}_{n\mathrm{\times }n}\mathit{}\mathrm{(}\mathrm{C}\mathrm{)}$.

Proof.

Define $\phi :R_n\to M_{n\times n}(ℂ)$ by for $A,B\in M_{n\times n}(ℝ)$.

Let $A,B,C,D\in M_{n\times n}(ℝ)$ such that . Then $A+iB=C+iD$. Therefore, $A=C$ and $B=D$. Hence, . It follows that $\phi$ is injective.

Let $Z\in M_{n\times n}(ℂ)$. Then there exist $X,Y\in M_{n\times n}(ℝ)$ such that $X+iY=Z$. Since , it follows that $\phi$ is surjective.

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

and

It follows that $\phi$ 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