Strang's book, question about exercise

  • Thread starter lukaszh
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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 [tex]z=a+\mathrm{i}b[/tex] is half of [tex]z+\overline{z}[/tex], and the real part of Z is half of [tex]Z+Z^H[/tex]."

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

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

But I can't understand that

[tex]\Re(Z)=\frac{1}{2}(Z+Z^H)[/tex]

if [tex]Z^H=(\overline{Z})^T[/tex]

Could you help me? Or tell me what is the real part of Z?
 
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Every complex matrix [tex]A[/tex] has a decomposition into a hermitian part

[tex]A_H=\frac{1}{2}(A+A^H)[/tex]

and a skew hermitian part

[tex]A_S=\frac{1}{2}(A-A^H)[/tex]

so that [tex]A=A_H+A_S[/tex]. 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.
 

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