×

The generalized spectral radius, numerical radius and spectral norm. (English) Zbl 0565.15009

Given an n by n matrix C with eigenvalues \(\gamma_ 1,...,\gamma_ n\), the generalized spectral radius \(\rho_ c\) and generalized numerical radius \(r_ c\) are defined for a matrix A by the formulae \(\rho_ c(A)=\max \{| \sum \alpha_ j\gamma_{\pi (j)}| \}\) as \(\pi\) ranges over all permutations of the indices 1,2,...,n, the \(\alpha\) being the eigenvalues of A.
Similarly \(r_ c(A)=\max \{| tr CUAU| \}\), \(| A|_ c=\max \{| tr CUAV| \}\) as U and V range over all unitary matrices. The authors characterize matrices for which two of these characteristics coincide. An example: (1) C is normal iff \(\rho_ c(A)=r_ c(A)\) for all normal A, (2) C is a multiple of the identity iff \(\rho_ c(A)=r_ c(A)\) for all A. The results generalize a number of classical results.
Reviewer: V.Pták

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Donoghue W. F., Michigan J. Math. 4 pp 261– (1957) · Zbl 0082.11601 · doi:10.1307/mmj/1028997958
[2] Furuta T., Proc. Japan. Acad 47 pp 279– (1971) · Zbl 0227.47002 · doi:10.3792/pja/1195520010
[3] Goldberg M., Linear Multilinear Algebra 7 pp 329– (1979) · Zbl 0416.15015 · doi:10.1080/03081087908817291
[4] Goldberg M., Linear Algebra Appl. 18 pp 1– (1977) · Zbl 0358.15005 · doi:10.1016/0024-3795(77)90075-1
[5] Goldberg M., Linear Algebra Appl. 24 pp 113– (1979) · Zbl 0395.15011 · doi:10.1016/0024-3795(79)90152-6
[6] Goldberg M., Linear Algebra Algebra 2 pp 317– (1975) · Zbl 0305.15004 · doi:10.1080/03081087508817075
[7] Goldberg M., Linear Algebra Appl 8 pp 427– (1974) · Zbl 0294.15010 · doi:10.1016/0024-3795(74)90076-7
[8] Horn A., Amer. J. Math 76 pp 620– (1954) · Zbl 0055.24601 · doi:10.2307/2372705
[9] Johnson C. R., Linear Algebra Appl. 15 pp 89– (1976) · Zbl 0337.15019 · doi:10.1016/0024-3795(76)90080-X
[10] Marcus M., Linear Algebra Appl. 21 pp 217– (1978) · Zbl 0393.15015 · doi:10.1016/0024-3795(78)90084-8
[11] Marcus M, Introduction to Linear Algebra (1965)
[12] Marcus M., Can. J. Math. 2 pp 419– (1978) · Zbl 0344.15016 · doi:10.4153/CJM-1978-036-6
[13] Von Neumann J., Tomsk Univ Rev. 1 pp 286– (1937)
[14] Pták V., Časopis pro Pěstovani Matematiky 101 pp 383– (1976)
[15] Rudin W., Functional Analysis (1973) · Zbl 0253.46001
[16] Thompson R. C., SIAM J Appl. Math. 32 pp 39– (1977) · Zbl 0361.15009 · doi:10.1137/0132003
[17] Wintner A., Math. Z. 39 pp 228– (1929) · JFM 55.0826.01 · doi:10.1007/BF01187766
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.