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set5 - $ Set 5 Symmetric Eigenvalue Problem Part 1 Kyle A...

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a39 a38 a36 a37 Set 5: Symmetric Eigenvalue Problem Part 1 Kyle A. Gallivan Department of Mathematics Florida State University Numerical Linear Algebra 1 Fall 2010 1
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a39 a38 a36 a37 Hermitian Eigenvalue Problem Theorem 5.1. If M C n × n is an Hermitian matrix then there exists a unitary matrix Q C n × n such that M = Q Λ Q H where Λ R n × n is a diagonal matrix with real entries. 2
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a39 a38 a36 a37 Hermitian Eigenvalue Problem Definition 5.1. The columns of Q are eigenvectors of M and the diagonal elements of Λ are the corresponding eigenvalues. MQ = Q Λ MQe i = Q Λ e i Mq i = q i λ i ( M λ i I ) q i = 0 det ( M λ i I ) = 0 3
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a39 a38 a36 a37 Hermitian Eigendecompostion An eigenvalue is a shift that makes M singular. The eigenvalues are the roots of the determinant of M λI . The eigenvectors are mutually orthogonal directions in C n along which the action of M is equivalent to scaling. The eigenvectors are unique up to scaling and for repeated eigenvalues up to the choice of basis for the associated eigenspace. If M is symmetric then Q R n × n is an orthogonal matrix. 4
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a39 a38 a36 a37 Basic Facts for Hermitian Matrices Reading GV96 Chapter 8.4 A = A H λ R , Aq = AQ = Q Λ , QQ H = Q H Q = I Λ = diag ( λ 1 , · · · , λ n ) A R n × n Q R n × n Generically, this cannot be computed in a finite number of computations. Similarity transformations do not change eigenvalues and change eigenvectors deterministically. 5
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a39 a38 a36 a37 Basic Idea for Jacobi Methods Jacobi methods produce Q as a convergent product of rotations. A 1 A, A k +1 = U k A k U T k , k = 1 , 2 , . . . The matrix U k is orthogonal and represents the action of the k -th sweep through the matrix. U k is the product of n ( n 1) / 2 plane rotations, i.e., a factored form of U k is produced. The particular choice of the elements and their ordering that define each rotation defines the “sweep” and therefore the method.
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