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February 7th, 2008, 09:24 PM  #1 
Newbie Joined: Feb 2008 Posts: 1 Thanks: 0  Need help with eigenvalue proof
Problem: Suppose we have a basis b_1, ..., b_n of eigenvectors of A, with corresponding eigenvalues a_i for 1 <= i <= n. Show that the product (A  a_1 * I) .... (A  a_n * I) = 0 Thank you in advance!! 
May 7th, 2008, 08:19 AM  #2 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7 
Well, by definition of an eigenvector, each b_i will be a root of the product (this product being a linear map). Since the set of all eigenvectors constitutes a basis of the vector space on which this linear mapping is defined, that simply means that the mapping must be identically 0 on all the space (since a linear mapping is uniquely defined by the values it takes on a basis).


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