On close eigenvalues of tridiagonal matrices |
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Authors: | Qiang >Ye |
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Institution: | (1) Department of Applied Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 e-mail: ye@newton.amath.umanitoba.ca , CA |
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Abstract: | Summary.
A symmetric tridiagonal matrix with a multiple eigenvalue must
have a zero
subdiagonal element and must be a direct sum of two
complementary blocks, both of which have the eigenvalue.
Yet it is well known that a small spectral gap
does not necessarily imply that some
is small, as
is demonstrated by the Wilkinson matrix.
In this note, it is shown that a pair of
close eigenvalues can only arise from two
complementary blocks on the diagonal,
in spite of the fact that the
coupling the
two blocks may not be small.
In particular, some explanatory bounds are derived and a
connection to
the Lanczos algorithm is observed. The nonsymmetric problem
is also included.
Received
April 8, 1992 / Revised version received September 21,
1994 |
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Keywords: | Mathematics Subject Classification (1991):65F15 15A42 |
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