Matrix Root

An th matrix root of a square matrix is a matrix satisfying

(1)

where is an integer and the matrix power uses matrix multiplication. The case is a matrix square root. A matrix root is not necessarily unique, and existence depends on both the matrix and the field of allowed entries (Higham 2008).

Every matrix root and its matrix are commuting matrices, since

(2)

Suppose an complex matrix has distinct nonzero eigenvalues, and write its eigen decomposition as , where the diagonal matrix is . Commutation forces every matrix root to preserve each one-dimensional eigenspace. Consequently, all its th roots have the form

(3)

where . The independent choices at each eigenvalue give exactly roots. Repeated or zero eigenvalues require separate consideration and may change the number of roots. For example, the identity matrix has infinitely many matrix square roots, while the nonzero nilpotent matrix with a single Jordan block has none.

Choudhry (2004) gives algebraic formulas for all th roots of matrices for , including singular matrices and cases with repeated eigenvalues.