**Abstract. **
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, 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.

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