Deturck, Wilf - Lecture Notes on Numerical Analysis (523142), страница 29
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Then Ax = λx. If we change basis, A changes toB = HAH −1 , or A = H −1 BH. Hence H −1 BHx = λx, or B(Hx) = λ(Hx). Therefore Hx3.12 Remarks125is an eigenvector of B with the same eigenvalue λ. The eigenvalues are therefore independentof the basis, and are properties of the linear mapping T itself. Hence we can speak of theeigenvalues of a linear mapping T .In the Jacobi method we carry out transformations A → JAJ T , where J T = J −1 . Hencethis transformation corresponds exactly to looking at the underlying linear mapping in adifferent basis, in which the matrix that represents it is a little more diagonal than before.Since J is an orthogonal matrix, it preserves lengths and angles because it preserves innerproducts between vectors:hJx, Jyi = hx, J T Jyi = hx, yi.(3.12.2)Therefore the method of Jacobi works by rotating a basis slightly, into a new basis in whichthe matrix is closer to being diagonal..
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