Extension and further development of the differential calculus for matrix norms with applications

In this paper, the differential calculus for the operator norms ||·||p, p∈{1,2,∞}, of the fundamental matrix or evolution Φ(t)=eAt, t⩾0, of a complex n×n matrix A, introduced by the author in a former paper, is extended to m times continuously differentiable matrix functions χ(t), t⩾0, and developed further for other p-norms |·|p, 1<p<∞. Results similar to those for Φ(t) are obtained. In addition, for this function Φ(t), formulae for the first two logarithmic derivatives D+1|Φ(0)|p and D+2|Φ(0)|p, 1<p<∞, are obtained as special cases. Also, upper bounds on the discrete evolution Ψ(t), t⩾0 (that is, a matrix power function) and on the difference (or remainder) R(t)=Φ(t)−Ψ(t), t⩾0, are derived. The discrete evolution occurs when a step-by-step method is employed to approximate the exact solution of the initial-value problem ẋ(t)=A x(t), x(0)=x0, which here models a vibration problem. The results are applied to the computation of the optimal upper bounds on ||R(t)||∞, ||R(t)||2, and |R(t)|2.