Matrix Square Root

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

(1)

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.

The family of real symmetric matrices

(2)

satisfies for every real , where is the identity matrix (Potts 1976, Higham 2020).

The matrix

(3)

has exactly four square roots over the complex numbers (Trott 2006). They can be found by solving the polynomial equations using a Gröbner basis. Its three distinct eigenvalues are 0 and , where and . Each square root preserves the one-dimensional eigenspaces and independently takes the values and on the two nonzero eigenspaces. It is zero on the null space. None of these roots is a real matrix, since its restriction to the real eigenspace for the simple negative eigenvalue would be a real scalar whose square is negative.

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

(4)

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].