I have some complex vectors that I would like to normalize and further manipulate. For example, one is this:
${\qquad \rm vec}=(-\frac{-a + b + \sqrt{a^2 - 2 a b + b^2 + 4 c^2 + 4 d^2}}{2 (c + i d)},1)$
where I have declared $a,b,c,d$ to be real.
I normalized it simply by doing FullSimplify[Normalize[vec]]
, which resulted in the following normalization factor:
$\qquad \frac{1}{2}\sqrt{4 + {\rm Abs}(\frac{-a + b +\sqrt{(a - b)^2 + 4 (c^2 + d^2)}}{c + i d})^2}$
I do not want the absolute value in this expression. I want to to be:
$\qquad \frac{1}{2}\sqrt{4 + \frac{(-a + b +\sqrt{(a - b)^2 + 4 (c^2 + d^2)})^2}{c^2 + d^2}}$
My first question: is there an elegant way to produce this normalization factor?
There is an additional complication. If I hard-code in this normalization factor, call it normFactor
, and attempt to take the conjugate of my normalized vector like so:
FullSimplify[Conjugate[vec/normFactor]]
The result has Conjugate
in it. For example one of the vector components is
$\qquad 2{\rm Conjugate}\frac{1}{\sqrt{4+\frac{(-a+b+\sqrt{(a - b)^2 + 4 (c^2 + d^2)})^2}{c^2 + d^2}}}$
even though the term under the square roots are all positive because I have declared $a,b,c,d$ to be real.
My second and third questions: why is Conjugate
behaving this way, and how can I fix it?
FullSimplify[Norm[...]/. Abs -> Identity]
$\endgroup$