Lecture #10 Notes

Then y y u xu u xu u x u u xu u x xu

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Unformatted text preview: t A∞ is normal. From last lecture’s proposition of QR algorithm we know that An ’s are all unitarily equivalent. It can be checked that any matrix unitarily equivalent to a normal matrix is also normal: say X and Y are unitarily equivalent, and X is normal. Then, Y ∗ Y = (U ∗ XU )∗ (U ∗ XU ) = U ∗ X ∗ U U ∗ XU = U ∗ X ∗ XU = U ∗ XX ∗ U = U ∗ XU U ∗ X ∗ U = Y Y ∗, and thus Y is normal. Hence, since An ’s are unitarily equivalent to A0 = A, and A is normal, and thus the limiting matrix A∞ is also normal.) By normality of R∞ and the fact that it is triangular, R∞ = D∞ with D∞ a diagonal matrix and A ∞ = D∞ Q ∞ = D∞ Q ∞ . Since {Q∞ , D∞ } is a commuting family in Mn , there exists a unitary U ∈ Mn that diagonalizes both: ￿ U ∗ D∞ U = D∞ , and U ∗ Q∞ U = Q∞ , ￿ ￿ where Q∞ is a diagonal matrix. We will denote the i’th diagonal entry of of Q∞ by ωi . Note that |ωi | = 1, since the eigenvalues of an unitary matrix are ±1, and the eigenvalues of Q∞ are ￿ the diagonal entries of Q∞ . Hence, we obtain the following result: U ∗ D∞ Q∞ U = U ∗ D∞ U U ∗ Q∞ U ￿ = D∞ Q ∞ d11 0 . . . 0 ω1 0 . . . 0 . . . . . . 0 d22 . . . 0 ω2 . . . = . . . . .. .. .. .. . . 0 . . 0 . . 0 . . . 0 dnn 0 . . . 0 ωn d11 ω1 0 ... 0 . . . d22 ω2 . . . 0 = . . .. .. . . ....
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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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