Abstract
In this chapter we study the influence of perturbations of matrices on the solutions of such basic problems of linear algebra as calculating eigenvalues and singular values of operators, constructing the inverse matrix, solving systems of linear algebraic equations, and solving the linear least squares problem.
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Notes
- 1.
For \(k=1\) and \(k=n\), these inequalities are modified in the obvious way.
- 2.
Hint: for a given x, write \(\Vert r(x,\alpha )\Vert _2^2\) as a quadratic trinomial in \(\alpha \).
- 3.
See the footnote on p. 232.
- 4.
Semyon Aronovich Gershgorin (1901–1933) was a Soviet mathematician.
- 5.
Friedrich Ludwig Bauer (1924–2015) was a German mathematician. Charles Theodore Fike (born 1933) is an American mathematician.
- 6.
By \(\tilde{b}_{jk}\) we denote the jth element of the column \(\tilde{b}_k\).
- 7.
See the notation in Section 7.3.
- 8.
See the proof of Theorem 7.11.
- 9.
Robert D. Skeel (born 1947) is an American mathematician.
- 10.
- 11.
Per-Ã…ke Wedin (born 1938) is a Swedish mathematician.
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Beilina, L., Karchevskii, E., Karchevskii, M. (2017). Elements of Perturbation Theory. In: Numerical Linear Algebra: Theory and Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-57304-5_7
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DOI: https://doi.org/10.1007/978-3-319-57304-5_7
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