sqrtM

sqrtm is the matrix square root function, defined for square matrices A over the real or complex field. A matrix X is a square root of A if X^2 = A. In general, a matrix can have more than one square root; there is often a distinguished or principal square root used in numerical work.

The principal square root is defined when appropriate by the holomorphic functional calculus: if A has no eigenvalues on the nonpositive real axis and a suitable branch of the complex square root is chosen, then there is a unique principal square root, denoted sqrtm(A), whose eigenvalues lie in the closed right half-plane. For Hermitian positive semidefinite A, there is a unique Hermitian positive semidefinite square root, which is widely used in applications.

Existence and uniqueness statements depend on the spectral structure of A. In particular, every nonsingular complex matrix has at least one square root, and the principal square root is well-defined under the standard spectral conditions. For singular matrices, a square root may or may not exist depending on the Jordan structure associated with the eigenvalue zero.

Numerical computation of sqrtm commonly relies on several approaches. The Schur decomposition A = Q T Q* allows a straightforward computation of sqrtm(A) from sqrtm(T) by X = Q sqrtm(T) Q*, with sqrtm(T) obtained recursively. Iterative methods such as Denman–Beavers and Newton–Schulz update pairs of matrices toward a square root, often with scaling to improve convergence and stability. Pade approximants provide another route, particularly within matrix function libraries.

Applications appear in control theory, differential equations, quantum mechanics, and graphics, wherever a matrix functional calculus or a square-root transformation of a linear operator is needed. See also matrix functions, Lyapunov equations, and the matrix exponential.