Hyyrö, Seppo; Merikoski, Jorma Kaarlo; Virtanen, Ari Improving certain simple eigenvalue bounds. (English) Zbl 0606.15011 Math. Proc. Camb. Philos. Soc. 99, 507-518 (1986). 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 Cited in 1 Document MSC: 15A42 Inequalities involving eigenvalues and eigenvectors Keywords:eigenvalue bounds; improvement of bounds; Rayleigh quotient; extreme eigenvalues; Perron root PDFBibTeX XMLCite \textit{S. Hyyrö} et al., Math. Proc. Camb. Philos. Soc. 99, 507--518 (1986; Zbl 0606.15011) Full Text: DOI References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.