Tridiagonal normal forms for orthogonal similarity classes of symmetric matrices*1
نویسندگان
چکیده
منابع مشابه
Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices
In this paper we present an O(nk) procedure, Algorithm MR3, for computing k eigenvectors of an n× n symmetric tridiagonal matrix T . A salient feature of the algorithm is that a number of different LDLt products (L unit lower triangular, D diagonal) are computed. In exact arithmetic each LDLt is a factorization of a translate of T . We call the various LDLt products representations (of T ) and,...
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Proof. We use induction on n. For n = 1, the result is immediate, First, observe there exist linear transformations Ui (i = 1, 2): X + X such that (U,x, y) = &(x, y). For define&: X -R by Liz(y) = #Q(x, y). Then L,, is a linear functional and since X is an inner product space, there exists Z< E X with L,,(y) = (zi, y) by the canonical isomorphism between X and its dual. Define the transformatio...
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Suppose that one knows an accurate approximation to an eigenvalue of a real symmetric tridiagonal matrix. A variant of deflation by the Givens rotations is proposed in order to split off the approximated eigenvalue. Such a deflation can be used instead of inverse iteration to compute the corresponding eigenvector. © 2002 Elsevier Science Inc. All rights reserved.
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The LBL factorization of Bunch for solving linear systems involving a symmetric indefinite tridiagonal matrix T is a stable, efficient method. It computes a unit lower triangular matrix L and a block 1×1 and 2×2 matrix B such that T = LBL . Choosing the pivot size requires knowing a priori the largest element σ of T in magnitude. In some applications, it is required to factor T as it is formed ...
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ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 2004
ISSN: 0024-3795
DOI: 10.1016/s0024-3795(04)00032-1