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Matrix shapes invariant under the symmetric QR algorithm
Author(s) -
Arbenz Peter,
Golub Gene H.
Publication year - 1995
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1680020203
Subject(s) - tridiagonal matrix , mathematics , algorithm , matrix (chemical analysis) , eigenvalues and eigenvectors , symmetric matrix , band matrix , qr decomposition , cuthill–mckee algorithm , invariant (physics) , algebra over a field , square matrix , pure mathematics , physics , materials science , quantum mechanics , composite material , mathematical physics
The QR algorithm is a basic algorithm for computing the eigenvalues of dense matrices. For efficiency reasons it is prerequisite that the algorithm is applied only after the original matrix has been reduced to a matrix of a particular shape, most notably Hessenberg and tridiagonal, which is preserved during the iterative process. In certain circumstances a reduction to another matrix shape may be advantageous. In this paper, we identify which zero patterns of symmetric matrices are preserved under the QR algorithm.