Roots of Matrices in the Study of GMRES Convergence and Crouzeix's Conjecture

For any nonsingular matrix A and any positive integer m, the mth root, A1/m, can be defined using any branch cut of the mth root function that does not pass through an eigenvalue of A. Cleary, A1/m approaches the identity as m → ∞, but we are interested in how it approaches the identity. It is shown that limm → ∞ [W( A1/m ) ]ᵐ = exp [ W( log A ) ], where W denotes the field of values and log A is defined using the same branch cut. It is also shown that \| A1/m \|ᵐ approaches exp( α ( log A ) ), where \| · \| denotes the spectral norm and α ( · ) is the numerical abscissa. Implications for the convergence rate of the GMRES algorithm are discussed, especially when $W(A)$ contains the origin. Additionally, it is shown that if A is a strict contraction and ρ ∈ ( \| A \| , 1 ), then the matrices [ ( I + ρ A )⁻¹ (A + ρ I ) ]1/m, m=1,2, …, are all strict contractions. The significance of results of this sort in a possible approach to proving Crouzeix's conjecture is discussed.