6. Matrix Square Root
Society for Industrial and Applied Mathematics eBooks · 2008
The matrix square root is one of the most commonly occurring matrix functions, arising most frequently in the context of symmetric positive definite matrices. The key roles that the square root plays in, for example, the matrix sign function (Chapter 5), the definite generalized eigenvalue problem (page 35), the polar decomposition (Section 2.6 and Chapter 8), and the geometric mean (Section 2.10), make it a useful theoretical and computational tool. The rich variety of methods for computing the matrix square root, with their widely differing numerical stability properties, are an interesting subject of study in their own right. We will almost exclusively be concerned with the principal square root, A1/2. Recall from Theorem 1.29 that for A ∈ ℂn×n with no eigenvalues on ℝ−, A1/2 is the unique square root X of A whose spectrum lies in the open right half-plane. We will denote by an arbitrary, possibly nonprincipal square root. We note the integral representation A1/2 = 2 π A ∫ 0 ∞ ( t2 I+A)−1 dt, 6.1 which is a special case of (7.1) in the next chapter. The integral can be deduced from that for the matrix sign function (see Problem 6.1). This chapter begins with analysis of the conditioning of the matrix square root and the sensitivity of the relative residual. Then a Schur method, and a version working entirely in real arithmetic, are described. Newton's method and several variants follow, with a stability analysis revealing that the variants do not suffer the instability that vitiates the Newton iteration. After a discussion of scaling, numerical experiments are given to provide insight into the analysis. A class of coupled iterations obtained via iterations for the matrix sign function are derived and their stability proved. Linearly convergent iterations for matrices that are “almost diagonal”, as well as for M-matrices, are analyzed, and a preferred iteration for Hermitian positive definite matrices is given. The issue of choosing from among the many square roots of a given matrix is addressed by considering how to compute a small-normed square root. A brief comparison of the competing methods is given. Finally, applications of involutory matrices, and some particular involutory matrices with explicit representations, are described.