EDB — 1MG

Exercises

1. [1MG] Prerequisites:Section . [2CM]↺↻,[118]↺↻, [11J]↺↻, [11F]↺↻, [1M5]↺↻.

   We equip the space of the matrices \(ℂ^{n× n}\) with one of the norms seen in Section [2CN]↺↻.

   • Show that the series \(∑_{k=0}^∞{A^ k}/{k!}\) converges.

   • Show that

     \begin{equation} \exp (A)=\lim _{N→∞}\Big({\mathbb {I}}+A/N\Big)^ N \label{eq:exp(A)=lim(I+A/N)N} \end{equation}

     19

     where \({\mathbb {I}}\) is the identity matrix in \(ℝ^{n× n}\); and that convergence is uniform in every compact neighborhood of \(A\). (Hint: make good use of the similar result [1M5]↺↻.)

   Solution 1

   [1MH]↺↻