Functions Preserving Matrix Groups and Iterations for the Matrix Square Root

Abstract

For which functions f does $A\in\mathbb{G} \Rightarrow f(A)\in\mathbb{G}$ when $\mathbb{G}$ 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 $f(A^{-1}) = f(A)^{-1}$ 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 $f(\overline{A}) = \overline{f(A)}$, and analytic functions and rational functions satisfying this condition are also characterized. These results enable us to characterize all meromorphic functions that map every $\mathbb{G}$ 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 $\mathbb{G}$. 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 $\mathbb{G}$; 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 $A\in\mathbb{G}$ 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 $A^{1/2}$ when $A\in\mathbb{G}$. Our tools include a formula for the sign of a certain block $2\times 2$ 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 $A\in\mathbb{G}$.