On localization of the eigenvalues of matrices "close" to triangular ones

Yıl 2022, Cilt: 51 Sayı: 4, 1104 - 1107, 01.08.2022

Michael Gil'

https://doi.org/10.15672/hujms.995747

Öz

We suggest a new bound for the eigenvalues of a matrix. For matrices which are "close" to triangular ones that bound is sharper than the well-known results, such as the Ostrowski theorem.

Anahtar Kelimeler

matrices, localization of eigenvalues, Ostrowski theorem

Kaynakça

• [1] A. Brauer, Limits for the characteristic roots of a matrix. II: Applications to stochastic matrices, Duke Math. J. 14 (1), 21-26, 1947.
• [2] M. Fiedler, F.J. Hall and R. Marsli, Gershgorin discs revisited, Linear Algebra Appl. 438 (1), 598-603, 2013.
• [3] S.A. Gershgorin. Uber die abgrenzung der eigenwerte einer matrix, Bull. Acad. des Sci. URSS 6, 749-754, 1931.
• [4] M.I. Gil, Perturbations of determinants of matrices, Linear Algebra and its Appl. 590, 235–242, 2020.
• [5] Ch.R. Johnson, J.M. Peña and T. Szulc, Optimal Gershgorin style estimation of the largest singular value, II, Electron. J. Linear Algebra, 31, 679-685, 2016.
• [6] C.K. Li and F. Zhang, Eigenvalue continuity and Gershgorin’s theorem, Electron. J. Linear Algebra 35, 619-625, 2019.
• [7] M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston 1964.
• [8] S. Milicević, V.R. Kostić, Lj. Cvetković and A. Miedlar, An implicit algorithm for computing the minimal Gershgorin set, Filomat, 33 (13), 4229-4238, 2019.
• [9] A. Ostrowski, Uber die determinanten mit űberwiegender hauptdiagonale, Comment. Math. Helv. 10, 69-96, 1937.
• [10] A. Ostrowski. Uber das nichtverschwinden einer klasse von determinanten und die lokalisierung der charakteristischen wurzeln von matrizen, Compositio Mathematica, 9, 209–226, 1951.