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Improving certain simple eigenvalue bounds. (English) Zbl 0606.15011

For a real symmetric matrix A, for \(e=(1,...,1)^ T\) and \(x=A^{\ell}e\) a property of the Rayleigh quotient is that \(\mu_{k\ell}(A)=(x^ TA^ kx)/x^ Tx\) gives bounds for the extreme eigenvalues. Some of the \(\mu_{k\ell}(A)'s\) are compared with each other. Relations between some of these bounds and a lower bound \(\mu\) (A) for the Perron root given by E. Deutsch are shown. For certain \(\mu_{k\ell}(A)\) and for \(\mu\) (A) the improvement is studied which can be achieved by varying the parameter t in \(\mu_{k\ell}(A-tI)+t\).
Reviewer: A.Bunse-Gerstner

MSC:

15A42 Inequalities involving eigenvalues and eigenvectors
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[1] DOI: 10.1080/03081088508817671 · Zbl 0589.15013 · doi:10.1080/03081088508817671
[2] Deutsch, Math. Proc. Cambridge Philos. Soc 92 pp 49– (1982)
[3] Berman, Nonnegative Matrices in the Mathematical Sciences (1979) · Zbl 0484.15016
[4] DOI: 10.1093/qmath/11.1.137 · Zbl 0136.24905 · doi:10.1093/qmath/11.1.137
[5] DOI: 10.1016/0024-3795(80)90258-X · Zbl 0435.15015 · doi:10.1016/0024-3795(80)90258-X
[6] DOI: 10.1007/BF01386332 · Zbl 0107.34305 · doi:10.1007/BF01386332
[7] DOI: 10.1215/S0012-7094-66-03360-6 · Zbl 0166.03901 · doi:10.1215/S0012-7094-66-03360-6
[8] Mirsky, Mathematika 3 pp 127– (1956)
[9] DOI: 10.2307/2309342 · Zbl 0094.00903 · doi:10.2307/2309342
[10] DOI: 10.1007/BF01931220 · Zbl 0398.65016 · doi:10.1007/BF01931220
[11] Marcus, Pacific J. Math 12 pp 627– (1962) · Zbl 0111.01502 · doi:10.2140/pjm.1962.12.627
[12] London, Pacific J. Math 16 pp 515– (1966) · Zbl 0136.25001 · doi:10.2140/pjm.1966.16.515
[13] DOI: 10.1215/S0012-7094-48-01560-9 · Zbl 0031.14803 · doi:10.1215/S0012-7094-48-01560-9
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