Ackoo

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!!

julien

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).