Lecture #10 Notes

# Thus q is diagonal and unitary and d11 1 0 0 d22

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Unformatted text preview: . 0 0 ... 0 dnn ωn Since the matrix D∞ Q∞ commutes with A∞ , and A∞ is unitarily equivalent to A (follows from the QR algorithm), D∞ Q∞ is unitarily equivalent to A. So λj = djj ωj are the eigenvalues of A. Thus, we know |djj | = |λj |. Moreover, if the diagonal entries of D∞ are distinct (magnitudes of the eigenvalues of A are distinct), and since D∞ Q∞ = Q∞ D∞ , the oﬀ diagonal entries of Q∞ ￿ must be zeros, and we precisely have Q∞ = Q∞ . Thus, Q∞ is diagonal and unitary, and d11 ω1 0 ... 0 . . . d22 ω2 . . . 0 A∞ = . . ... ... . . 0 0 ... 0 dnn ωn 2 So if the magnitudes of a normal matrix A are distinct and if the QR algorithm converges, then it diagonalizes A. 1 Real Matrices (cont’d) We proceed to show under which conditions a matrix with entries over R can be diagonalized the same way as matrices with complex entries. 1.5.1 Deﬁnition. A matrix A ∈ Mn (R) is similar to B ∈ Mn (R) if there exists invertible S ∈ Mn (R) such that B = S −1 AS. The matrix A is diagonalizable if A is similar to a diagonal matrix. 1.5.2 Theorem. A ∈ Mn (R) is diagonalizable if and only if there is a set of n linearly independent eigenvectors. Proof. As before, S −1 AS = D, with D diagonal implies that S = [x1 , x2 , · · · , xn ] contains a basis of n eigenvectors and vice versa. 1.5.3 Theo...
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## This note was uploaded on 01/16/2014 for the course MATH 6304 taught by Professor Bernhardbodmann during the Fall '12 term at University of Houston.

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