Lecture2B

# Semisimple nilpotent decomposition shift notation

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Unformatted text preview: r and refer to matrix instead. Let or and . Let be the diagonal matrix with the eigenvalues of repeated according to multiplicity. Let be a basis for of generalized eigenvectors of . Consider the transformation matrix and define . is a semisimple matrix. Multiply by on the right to obtain this result is , where are the distinct eigenvalues of as the diagonalizable part of . Consider an arbitrary vectors for : Then . The i^th component of and . Think of can be expressed as a linear combination of the basis . We then have . Within , acts as a multiple of the identity operator. In particular, action of . Lemma 2.7 Let with order at most Remark 1. is invariant under the , where . Then commutes with , the maximum of the algebraic multiplicities of . is nilpotent of order means the same thing as Remark 2. Since the generalized eigenspace , it is also invariant under the action of . of and is nilpotent has nilpotency . is invariant under the action of both Proof. See the text; plan to give it in class. Not...
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