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.