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Unformatted text preview: Jordan Normal Form April 24, 2007 Definition: A Jordan block is a square matrix B whose diagonal entries consist of a single scalar , whose superdiagonal entires are all 1, and all of whose other entries vanish. For example: 1 1 1 Theorem: Let T be a linear operator on a finite dimensional vector space V . Suppose that the characteristic polynomial of V splits. Then there exists a basis for T such that [ T ] is a direct sum of Jordan blocks. The first step in the proof of this theorem is to use the direct sum decom- position of V into generalized eigenspaces K . Then it suffices to prove the theorem for the restriction of T to each K . On K , let S := T- I . If we can find a basis of K with respect to which S is a sum of Jordan blocks, then the same will be true for T . On K , there exists an r such that S r = 0. Thus it suffices to consider the special case of operators with this property. Let V be a finite dimensional vector space over a field F . A linear operator N : V V is said to be nilpotent if N r = 0 for some positive integer r . Let N be a nilpotent operator on a finite dimensional vector space V . For each i , let R i be the image of N i . Each R i is a linear subspace of V and is N-invariant, and 0 = R r R r- 1 R 1 V . Since N is nilpotent it is not injective (unless V = 0). Thus the kernel K of N is not zero and dim R 1 = dim V- dim K < dim V ....
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This note was uploaded on 12/05/2010 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at University of California, Berkeley.
- Spring '08