The matrix square root from a new functional perspective: Theoretical results and computational issues

We give a new characterization of the matrix square root and a new algorithm for its computation. We show how the matrix square root is related to the constant block coefficient of the inverse of a suitable matrix Laurent polynomial. This fact, besides giving a new interpretation of the matrix square root, allows one to design an efficient algorithm for its computation. The algorithm, which is mathematically equivalent to Newton's method, is quadratically convergent and numerically insensitive to the ill-conditioning of the original matrix and works also in the special case where the original matrix is singular and has a square root.