I am trying to read Donaldson's paper on symplectic submanifolds

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214459407

and am getting a bit stuck on some simple linear algebra at the beginning. On the fourth page of the paper (p. 669) there is the following assertion: let $A: \mathbb{C}^n \rightarrow \mathbb{C}$ be an $\mathbb{R}$-linear transformation and write $A$ as the sum of its linear and antilinear parts $a'$ and $a''$. Then

$A$ has real rank 2 unless $\bar{a''} = e^{i\alpha}a'$ for some $\alpha$

and

the tangent of the Kahler angle is given by $2\sqrt{|a'|^2||a''|^2 - |\langle a', a'' \rangle|^2}/(|a'|^2 - |a''|^2)$.

Donaldson says this is evident upon "a little calculation", but it seems to be eluding me (sorry if the answer is simple).

If someone could give a bit more detailed derivation of these two statements that would be very helpful. Thanks!

I am trying to read Donaldson's paper on symplectic submanifolds

Link

and am getting a bit stuck on some simple linear algebra at the beginning. On the fourth page of the paper (p. 669) there is the following assertion: let $A: \mathbb{C}^n \rightarrow \mathbb{C}$ be an $\mathbb{R}$-linear transformation and write $A$ as the sum of its linear and antilinear parts $a'$ and $a''$. Then

$A$ has real rank 2 unless $\bar{a''} = e^{i\alpha}a'$ for some $\alpha$

and

the tangent of the Kahler angle is given by $2\sqrt{|a'|^2||a''|^2 - |\langle a', a'' \rangle|^2}/(|a'|^2 - |a''|^2)$.

Donaldson says this is evident upon "a little calculation", but it seems to be eluding me (sorry if the answer is simple).

If someone could give a bit more detailed derivation of these two statements that would be very helpful. Thanks!