Overview

Unfortunately, the roundoff errors in the mth power of a matrix, say B ^m, are usually small relative to B ^m rather than B ^m.

- CLEVE B. MOLER and CHARLES F. VAN LOAN, Nineteen Dubious Ways to Compute the Exponential of a Matrix (1978)

It is the size of the hump that matters: the behavior of p( A ? t) ⁿ = p( A ? t) ^{t/ ? t} for small but nonzero t. The eigenvalues and the norm, by contrast, give sharp information only about the limits t ? ? or t ? 0.

- DESMOND J. HIGHAM and LLOYD N. TREFETHEN, Stiffness of ODEs (1993)

Many people will go through life powering matrices and never see anything as dramatic as J ₂(0.99) ^k. ^[16]

- G. W. STEWART, Matrix Algorithms. Volume II: Eigensystems (2001)

Overview

Powers of matrices occur in many areas of numerical analysis. One approach to proving convergence of multistep methods for solving differential equations is to show that a certain parameter-dependent matrix is uniformly "power bounded" [537, 1991, V.7], [974, 1992]. Stationary iterative methods for solving linear equations converge precisely when the powers of the iteration matrix converge to zero. And the power method for computing the largest eigenvalue of a matrix computes the action of powers of the matrix on a vector. It is therefore important to understand the...