isomorphism of rings of real and complex matrices

Note that submatrix notation ( will be used within this entry. Also, for any positive integer n, Mn×n(R) will be used to denote the ring of n×n matrices with entries from the ring R, and Rn will be used to denote the following subring of M2n×2n():

Rn={PM2n×2n():P=(AB-BA) for some A,BMn×n()}

For any positive integer n, RnMn×n(C).


Define φ:RnMn×n() by φ((AB-BA))=A+iB for A,BMn×n().

Let A,B,C,DMn×n() such that φ((AB-BA))=φ((CD-DC)). Then A+iB=C+iD. Therefore, A=C and B=D. Hence, (AB-BA)=(CD-DC). It follows that φ is injectivePlanetmathPlanetmath.

Let ZMn×n(). Then there exist X,YMn×n() such that X+iY=Z. Since φ((XY-YX))=X+iY=Z, it follows that φ is surjectivePlanetmathPlanetmath.

Let A,B,C,DMn×n(). Then




It follows that φ is an isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( ∎

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