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Exercise 7.2.22

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Since $A$ is a matrix over complex numbers, $A$ has the Jordan canonical form $J = Q^{- 1} AQ$ for some invertible matrix $Q$ . So $e^{A}$ exists if $e^{J}$ exist. Observe that

∥ J^{m} ∥ \leq n^{m - 1} ∥ J ∥

by Exercise 7.2.20(d). This means the sequence in the definition of $e^{J}$ converge absolutely. Hence $e^{J}$ exists.

jephianlin

2011-06-27 00:00

Exercise 7.2.22

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