The triangular factorization of symmetric matrices, usually ascribed to A. L. Cholesky, was (essentially) explicitly given by O. Toeplitz earlier, but rather parenthetically. Actually, the word 'matrix' does n...
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The triangular factorization of symmetric matrices, usually ascribed to A. L. Cholesky, was (essentially) explicitly given by O. Toeplitz earlier, but rather parenthetically. Actually, the word 'matrix' does not appear in the report on Cholesky's process;the matrix formulation seems due to Henry Jensen.
Real nonsymmetric tridiagonal matrices arise in various applications. When one is asked to find the eigenvalues of such a matrix, the QR algorithm is used, but this destroys tridiagonal form by converting the matrix t...
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Real nonsymmetric tridiagonal matrices arise in various applications. When one is asked to find the eigenvalues of such a matrix, the QR algorithm is used, but this destroys tridiagonal form by converting the matrix to Hessenberg form, resulting in increased storage requirements and numerical operations. The HR algorithm, based on the HR factorization of the matrix into a (Delta, Delta(1))-orthogonal part H, where H(T) Delta H = Delta(1), and an upper triangular part R, solves this problem. In a result proved by Hongguo Xu, two steps of the LR algorithm are equivalent to one step of the QR algorithm for symmetric matrices. The first object of this paper is to use the HR algorithm to extend Hongguo Xu's result to the nonsymmetric case. Since an HR factorization does not always exist, so we also consider an extension to it called XHR factorization. We then prove a similar result about it.
Real nonsymmetric tridiagonal matrices arise in various applications. When one is asked to find the eigenvalues of such a matrix, the QR algorithm is used, but this destroys tridiagonal form by converting the matrix t...
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Real nonsymmetric tridiagonal matrices arise in various applications. When one is asked to find the eigenvalues of such a matrix, the QR algorithm is used, but this destroys tridiagonal form by converting the matrix to Hessenberg form, resulting in increased storage requirements and numerical operations. The HR algorithm, based on the HR factorization of the matrix into a (Delta, Delta(1))-orthogonal part H, where H(T) Delta H = Delta(1), and an upper triangular part R, solves this problem. In a result proved by Hongguo Xu, two steps of the LR algorithm are equivalent to one step of the QR algorithm for symmetric matrices. The first object of this paper is to use the HR algorithm to extend Hongguo Xu's result to the nonsymmetric case. Since an HR factorization does not always exist, so we also consider an extension to it called XHR factorization. We then prove a similar result about it.
PLUS factorizations, or customizable triangular factorizations, of nonsingular matrices have found applications in source coding and computer graphics. However, there are still some open problems. In this paper, we pr...
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PLUS factorizations, or customizable triangular factorizations, of nonsingular matrices have found applications in source coding and computer graphics. However, there are still some open problems. In this paper, we present a new necessary condition and a sufficient condition for the existence of generic PLUS factorizations. (c) 2004 Elsevier Inc. All rights reserved.
This letter shows that the matrix structure with 2 X 2 Alamouti sub-blocks remains invariant under several nontrivial matrix operations, including matrix inversion, Schur complementation, Riccati recursion, triangular...
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This letter shows that the matrix structure with 2 X 2 Alamouti sub-blocks remains invariant under several nontrivial matrix operations, including matrix inversion, Schur complementation, Riccati recursion, triangular factorization, and QR factorization.
Customizable triangular factorizations of matrices find their applications in computer graphics and lossless transform coding. In this paper, we prove that any N x N nonsingular matrix A can be factorized into 3 trian...
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Customizable triangular factorizations of matrices find their applications in computer graphics and lossless transform coding. In this paper, we prove that any N x N nonsingular matrix A can be factorized into 3 triangular matrices, A=PLUS, where P is a permutation matrix. L is a unit lower triangular matrix, U is all upper triangular matrix of which the diagonal entries are customizable and can be given by all means as long as its determinant is equal to that of A up to a possible sign adjustment, and S is a unit lower triangular matrix of which all but N-1 off-diagonal elements are set zeros and the positions of those N-1 elements are also flexibly customizable, such as a single-row, a single-column, a bidiagonal matrix or other specially patterned matrices. A pseudo-permutation matrix, which is a simple unit upper triangular matrix with off-diagonal elements being 0, 1 or -1, can take the role of the permutation matrix P as well. In some cases, P may be the identity matrix. Besides PLUS, a customizable factorization also has other alternatives, LUSP, PSUL or SULP for lower S, and PULS, ULSP, PSLU, SLUP for upper S. (C) 2004 Elsevier Inc. All rights reserved.
In this paper,algorithms for determining the triangular factorization of Cauchy type matrices and their inverses are derived by using O( n^2) operations.
In this paper,algorithms for determining the triangular factorization of Cauchy type matrices and their inverses are derived by using O( n^2) operations.
For an Hermitian matrix the QR transform is diagonally similar to two steps of the LR transforms. Even for non-Hermitian matrices the QR transform may be written in rational form.
For an Hermitian matrix the QR transform is diagonally similar to two steps of the LR transforms. Even for non-Hermitian matrices the QR transform may be written in rational form.
The authors present a fast procedure for computing a ''modified'' triangular factorization of Hankel, quasi-Hankel (matrices congruent in a certain sense to Hankel matrices) and sign-modified quasi-Han...
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The authors present a fast procedure for computing a ''modified'' triangular factorization of Hankel, quasi-Hankel (matrices congruent in a certain sense to Hankel matrices) and sign-modified quasi-Hankel (products of quasi-Hankel and signature matrices) matrices. A fast procedure for computing inverse of Hankel and quasi-Hankel matrices is also presented. A modified triangular factorization is an LDL* factorization, where L is lower triangular with unit diagonal entries and D is a block diagonal matrix with possibly varying block sizes. Only matrices with all leading minors nonzero, often called strongly regular, will always have a purely diagonal and nonsingular D matrix. The matrices studied in this paper have diagonal blocks with a particular Hankel-(like) structure. The algorithms presented here are obtained by extending a generating function approach of Lev-Ari and Kailath for matrices with a generalized displacement structure. A particular application of the results is a fast method of computing the rank profile and inertia of the matrices involved.
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