Strang's book, question about exercise

[lukaszh]

Hello,
now I'm reading G.Strang's book Linear algebra and its Applications, chapter about Hermitian matrices and complex matrices. In one of the exercises, there's a sentence:

"The real part of z=a+\mathrm{i}b is half of z+\overline{z}, and the real part of Z is half of Z+Z^H."

I know that first part of sentence is undoubtedly truth, so

\frac{1}{2}(z+\overline{z})=\frac{1}{2}(a+\mathrm{i}b+a-\mathrm{i}b)=\frac{2a}{2}=a=\Re(z)

But I can't understand that

\Re(Z)=\frac{1}{2}(Z+Z^H)

if Z^H=(\overline{Z})^T

Could you help me? Or tell me what is the real part of Z?

 

[yyat]

Every complex matrix A has a decomposition into a hermitian part

A_H=\frac{1}{2}(A+A^H)

and a skew hermitian part

A_S=\frac{1}{2}(A-A^H)

so that A=A_H+A_S. This looks formaly similar to taking the real and imaginary parts of a complex number, which may be the reason Strang calls them "real" and "complex" parts.