# Is invariant under the action of both proof see the

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Unformatted text preview: e that there are two parts: (1) show and (2) show nilpotent. and 6 Chapter 2 part B Theorem 2.8. A matrix on a complex vector space where is semisimple, is nilpotent and . has a unique decomposition , Proof. Not in lecture. See text. The Exponential Let , then by lemma 2.7 where . Further, we have , and , where is the maximum algebraic multiplicity of the eigenvalues. Then, using the law of exponents for commuting matrices and the series definition of the exponential [1] This formula allows us to compute the exponential of an arbitrary matrix. Combine this result with the fundamental theorem to find an analytical solution for any linear system. Example. Solve the initial value problem with given and . By the fundamental theorem, . We need to compute . . The characteristic equation is . The root multiplicity 2. Then and has . Every matrix commutes with the identity matrix, so that . Then . Notice that . N has nilpotency 2. Then using [1] , . 7 Chapter 2 part B Notice that if the straight...
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