Consider a complex vector space V,

if I decompose a Postive definite Hermitian form (,) into real and imaginary part, i get $$(v,w)=\lbrace v,w \rbrace +[v,w]i$$ How can I show that whenever two such herimitian form have equal imaginary part, the two forms are equal?
Thanks in advance.

My thoughts: I try to make use of a n dimension complex vector space has the same basis as n dimension real vector space has...

  • 2
    $\begingroup$ This follows directly from the fact that $(x,y)=\Im(y,ix)-i\Im(y,x)$, assuming that the inner product is anti-linear in its first argument. $\endgroup$
    – user1551
    Mar 17, 2022 at 5:59
  • $\begingroup$ Sorry, what is that J-like symbol? $\endgroup$ Mar 17, 2022 at 6:00
  • $\begingroup$ It's an I, not J. It denotes the imaginary part of a complex number. If $z=a+ib$ where $a$ and $b$ are real, then $x=\Re(z)=\frac12(z+\overline{z})$ and $y=\Im(z)=\frac{1}{2i}(z-\overline{z})$. $\endgroup$
    – user1551
    Mar 17, 2022 at 6:03
  • $\begingroup$ Oh i see, is this fact means that the hermitian form can be solely expressed by its imaginary part. So naturally the two hermitian form with equal imaginary part are equal? Am i think it right? $\endgroup$ Mar 17, 2022 at 6:11
  • 1
    $\begingroup$ Yes and this is essentially how the polarisation identity is proved. $\endgroup$
    – user1551
    Mar 17, 2022 at 6:22


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