Functions Preserving Matrix Groups and Iterations for the Matrix Square Root

For which functions f does A ∈ G ⇒ f(A) ∈ G when G is the matrix automorphism group associated with a bilinear or sesquilinear form? For example, if A is symplectic when is f(A) symplectic? We show that group structure is preserved precisely when f(A) = f(A) for bilinear forms and when f(A) = f(A) for sesquilinear forms. Meromorphic functions that satisfy each of these conditions are characterized. Related to structure preservation is the condition f(A) = f(A), and analytic functions and rational functions satisfying this condition are also characterized. These results enable us to characterize all meromorphic functions that map every G into itself as the ratio of a polynomial and its “reversal”, up to a monomial factor and conjugation. The principal square root is an important example of a function that preserves every automorphism group G. By exploiting the matrix sign function, a new family of coupled iterations for the matrix square root is derived. Some of these iterations preserve every G; all of them are shown, via a novel Fréchet derivative-based analysis, to be numerically stable. A rewritten form of Newton’s method for the square root of A ∈ G is also derived. Unlike the original method, this new form has good numerical stability properties, and we argue that it is the iterative method of choice for computing A1/2 when A ∈ G. Our tools include a formula for the sign of a certain block 2× 2 matrix, the generalized polar decomposition along with a wide class of iterations for computing it, and a connection between the generalized polar decomposition of I + A and the square root of A ∈ G.