The inverse problem of geometric and golden means of positive definite matrices (Q869255)

Given positive definte matrices \(A\) and \(B\), the inverse problem of the geometric and golden means consists in finding positive definite matrices \(X\) and \(Y\) such that: (1) \(X*Y=A\), (2) \((X+X*(4Y-3X))/2=B\), where \(X*Y=X^1{}^/{}^2 (X{}^-{}^1{}^/{}^2 Y X{}^-{}^1{}^/{}^2){}^1{}^/{}^2 X{}^1{}^/{}^2\) defines the geometric mean of \(X\) and \(Y\). It is shown that system (1), (2) is solvable (respectively uniquely solvable) iff \(A \leqslant 3{}^1{}^/{}^2 B \leqslant 2A\) (respectively \(A \leqslant 3{}^1{}^/{}^2 B \leqslant 3{}^1{}^/{}^2 A\)).