Assume there are $3$ random variables $X$, $Y$ and $Z$ such that $\operatorname{corr}(X,Y) = 0.5$, $\operatorname{corr}(X,Z) = -0.5$. What is the exact range for $\operatorname{corr}(Y, Z)$?
My approach: I tried using the formula for correlation i.e.
$$\frac{E[XY]-E[X]E[Y]}{\operatorname{std}(X)\operatorname{std}(Y)}$$
and substituting the variables but I didn't reach anywhere. Can someone suggest a way? I was looking at various explanations of similar type problems on the net and found they have certain relation with semidefinite matrix. I have no idea what they are. If they are useful, I would encourage you to kindly leave a link too so that I can learn it. Thanks!
From your assumptions it follows that there exist, two random variables $X^\bot,X^{\bot\bot}$ that are both uncorrelated with $X$ and have variance one such that \begin{align} Y&=c_{12}\,X+\sqrt{1-c_{12}^2}\,X^\bot\\ Z&=c_{13}\,X+\sqrt{1-c_{13}^2}\,X^{\bot\bot}\\ \end{align} where $c_{12}=1/2$ and $c_{13}=-1/2\,$. Writing $$ \operatorname{corr}(X^\bot,X^{\bot\bot})=\rho $$ we have
$$ \operatorname{corr}(Y,Z)=c_{12}\,c_{13}+\sqrt{1-c_{12}^2}\,\sqrt{1-c_{13}^2}\,\rho\,. $$ The range of this correlation is obtained from $\rho\in[-1,1]$.