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Strang's book, question about exercise

  1. Mar 23, 2009 #1
    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?
     
  2. jcsd
  3. Mar 23, 2009 #2
    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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