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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
. Further, we have
, where is the maximum algebraic multiplicity of the eigenvalues.
Then, using the law of exponents for commuting matrices and the series definition of the
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
Every matrix commutes with the identity matrix, so that . Then .
Notice that . N has nilpotency 2. Then using 
Chapter 2 part B Notice that if the straight...
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- Fall '14