Solution of quadratic matrix equations by least-squares method

A least-square method is described for obtaining the solutions $A$ and $B$ for the matrix equations ${Y^2} - YD + S = 0$ and ${Y^2} - DY + S = 0$ respectively. No limitation is set on $A$, $B$, $D$, and $S$ except that they be square matrices. An illustrated example, including computing procedures, is discussed. The mathematical solutions presented should prove useful for solving the eigenvalue problem $\left ( {{\lambda ^2}I + \lambda D + S} \right )X = 0$, especially when the dimension of the matrices is large.