1. The problem statement, all variables and given/known data I need to show that any normal matrix can be expressed as the sum of two commuting self adjoint matrices 2. Relevant equations Normal matrix A: [itex][A,A^\dagger]=0[/itex] Self Adjoint matrix: [itex]B=B^\dagger[/itex] 3. The attempt at a solution A is a normal matrix. I assume I can write any matrix as a sum of two other matrices (no conditions imposed yet). So: [tex]A=B+C[/tex] But A is normal so [tex]AA^\dagger=A^\dagger A[/tex] Expanding the [itex]AA^\dagger[/itex] term we have [tex]AA^\dagger=(B+C)(B^\dagger +C^\dagger )[/tex] [tex]AA^\dagger=BB^\dagger +BC^\dagger+CB^\dagger+CC^\dagger[/tex] This will only be equal to [itex]A^\dagger A[/itex] if B and C are self adjoints and commute with each other. (I could write the other steps but I think you got the point). The problem is that I'm not sure if this proves that ANY normal matrix can be written like the sum of two self adjoint and commuting matrices. I'm not sure what this proves at all. This would still be valid if B=I and C=0, right? Also, I've read some proofs using complex numbers and I just don't see where that's coming from.