Let $A=\begin{pmatrix} a & c \\ 0 & b \end{pmatrix}$ where $a \neq b $ or $c \neq 0$. $a,b,c $ being complex numbers.
I have to find every $X \in T_2^+(C) $ such that $X^2=A $.
I thing I have proved A admits square root(s) if and only if $a \neq 0$ and $b \neq 0$.
But then I can't figure out the square roots of A. I get the following equations: (If $X=\begin{pmatrix} d & f\\ 0 & e \end{pmatrix}$)
$d^2=a $
$e^2=b $
$(d+e)f=c $
But I don't know anything about square roots in C so I'm stuck..
Thanks!
Write $a=r(\cos\alpha +i\sin\alpha)$ and then $d=r^{1/2}(\cos\alpha/2 +i\sin\alpha/2)$. Write $b=r(\cos\theta+i\sin\theta)$ and then $e=r^{1/2}(\cos\theta/2 +i\sin\theta/2)$. Finally after get $d$ and $e$ just replace at $f=c/(d+e)$ to get $f$.