I would like to rearrange the matrix equation $s=A'B^{-1}A$ into the form $A=f(s,B)$ (i.e., some function of $s$ and $B$), where s is scalar, $A$ is $n\times 1$, $A'$ is the transpose of $A$, and $B$ is $n\times n$. I need a general solution, so that I can plug in values for $s$ and $R$ and solve for $P$. How should I go about this?
In case it helps, the equation in its current form calculates the squared multiple correlation ($R^2$) for the regression equation $y=x_1+x_2+...x_n$, given that $s=R^2$, $A$ is an $n\times 1$ vector of population correlations between y and each $x$, and $R$ is a square matrix of population correlations among the predictors themselves (so it's symmetric with a diagonal of 1's). My goal is to supply the multiple correlation and a set of predictors with known relationships, and find the correlations each predictor $x$ must have with the dependent variable $y$.
Thanks for your help!
EDIT: Assuming $s > 0$ and $B$ is positive definite, take any nonzero vector $v$, let $t = v' B^{-1} v$ and let $A = (s/t)^{1/2} v$.