6 - Recap of lecture on 4/6 Key steps in showing any real...

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Recap of lecture on 4/6 Key steps in showing any real symmetric matrix A has a basis of eigenvectors (1) The eigenvalues of a real symmetric matrix are real. We will not prove this now. (2) Invariant subspaces Recall that W is invariant for A provided A ( W ) W . That is , Aw is in W whenever w is in W . Fact W is also invariant , because A is symmetric. Proof W consists of vectors u such that u w = 0 for all w in W . We must show that Au w = 0 for all w in W . We have Au w = u A T w = u Aw = 0 because Aw is in W . (3) Theorem If W is an invariant subspace for the symmetric n × n matrix A , then there is an eigenvector of A lying in W . The proof is in several steps. (a) Choose an orthonormal basis { u 1 ,...,u p } for W and choose an orthonor- mal basis { u p +1 ,...,u n } for W . Then α = { u 1 ,...,u n } is an orthonormal basis R n . Let ± denote the standard basis for R n . Let P = P ± α be the
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orthogonal matrix P = [ u 1 ,...,u n ] and recall that
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6 - Recap of lecture on 4/6 Key steps in showing any real...

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