Matrix Square Root

A matrix square root of a square matrix is a matrix satisfying

The square uses matrix multiplication, not entry-by-entry multiplication. A matrix can have no square root, finitely many square roots, or infinitely many square roots, even among complex matrices. For example, the nonzero nilpotent matrix has no square root, while the identity matrix has infinitely many.

If has no eigenvalues on the nonpositive real axis, it has a unique square root whose eigenvalues have positive real parts. This is its principal matrix square root, denoted . A real symmetric matrix that is positive semidefinite also has a unique real symmetric matrix square root that is positive semidefinite. If its eigen decomposition is with an orthogonal matrix, this root is

where the diagonal matrix contains the nonnegative square roots of the eigenvalues. Matrix square roots are computed in the Wolfram Language using MatrixPower[a, 1/2].