A further sufficient condition for the determinantal conjecture
- Authors: Shitov Y.N.1
-
Affiliations:
- Issue: Vol 89, No 4 (2025)
- Pages: 219-226
- Section: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/306781
- DOI: https://doi.org/10.4213/im9292
- ID: 306781
Cite item
Open Access
Access granted
Subscription Access
Abstract
Let $A$, $B$ be $n\times n$ normal matrices with eigenvalues $(a_1,…,a_n)$, $(b_1,…,b_n)$, respectively. We show that $\det(A+B)$ lies in the convex hull ofif all eigenvalues of $A$, $B$ are real, except for three eigenvalues of $B$.
About the authors
Yaroslav Nikolaevich Shitov
Author for correspondence.
Email: yaroslav-shitov@yandex.ru
Doctor of physico-mathematical sciences, no status
References
- N. Bebiano, J. K. Merikoski, J. da Providência, “On a conjecture of G. N. de Oliveira on determinants”, Linear Multilinear Algebra, 20:2 (1987), 167–170
- N. Bebiano, J. da Providência, “Some remarks on a conjecture of de Oliveira”, Linear Algebra Appl., 102 (1988), 241–246
- N. Bebiano, J. F. Queiro, “The determinant of the sum of two normal matrices with prescribed eigenvalues”, Linear Algebra Appl., 71 (1985), 23–28
- N. Bebiano, G. Soares, “Three observations on the determinantal range”, Linear Algebra Appl., 401 (2005), 211–220
- N. Bebiano, Yiu Tung Poon, J. da Providência, “On $C$-$det$ spectral and $C$-$det$-convex matrices”, Linear Multilinear Algebra, 23:4 (1988), 343–351
- N. Bebiano, A. Kovačec, J. da Providência, “The validity of the Marcus–de Oliveira conjecture for essentially Hermitian matrices”, Linear Algebra Appl., 197/198 (1994), 411–427
- J. da Providência, N. Bebiano, “Matrices satisfying a conjecture of G. N. de Oliveira on determinants”, Linear Algebra Appl., 78 (1986), 187–198
- S. W. Drury, “Essentially Hermitian matrices revisited”, Electron. J. Linear Algebra, 15 (2006), 285–296
- S. W. Drury, “OMC for scalar multiples of unitaries”, Linear Algebra Appl., 422:1 (2007), 318–325
- G. de Oliveira, “Normal matrices (research problem)”, Linear Multilinear Algebra, 12:2 (1982), 153–154
- M. Fiedler, “Bounds for the determinant of the sum of Hermitian matrices”, Proc. Amer. Math. Soc., 30 (1971), 27–31
- S. Fisk, “A very short proof of Cauchy's interlace theorem”, Amer. Math. Monthly, 112:2 (2005), 118
- Huajun Huang, “A survey of the Marcus–de Oliveira бonjecture”, Advances in algebra, SRAC 2017 (Mobile, AL, 2017), Springer Proc. Math. Stat., 277, Springer, Cham, 2019, 169–181
- A. Kovačec, “On a conjecture of Marcus and de Oliveira”, Linear Algebra Appl., 201 (1994), 91–97
- A. Kovačec, “The Marcus–de Oliveira conjecture, bilinear forms, and cones”, Linear Algebra Appl., 289:1-3 (1999), 243–259
- Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze, “Ranks and determinants of the sum of matrices from unitary orbits”, Linear Multilinear Algebra, 56:1-2 (2008), 105–130
- M. Marcus, “Derivations, Plücker relations, and the numerical range”, Indiana Univ. Math. J., 22:12 (1972/73), 1137–1149
- J. K. Merikoski, A. Virtanen, “Some notes on de Oliveira's determinantal conjecture”, Linear Algebra Appl., 121 (1989), 345–352
- J. K. Merikoski, A. Virtanen, “Some further notes on the Marcus–de Oliveira determinantal conjecture”, Linear Algebra Appl., 187 (1993), 259–262
- Q. I. Rahman, G. Schmeisser, Analytic theory of polynomials, London Math. Soc. Monogr. (N.S.), 26, The Clarendon Press, Oxford Univ. Press, Oxford, 2002, xiv+742 pp.
- K. Rodtes, “A class of normal dilation matrices affirming the Marcus–de Oliveira conjecture”, Oper. Matrices, 15:1 (2021), 127–130
- N. Bebiano, J. P. da Providência, “Revisiting the Marcus–de Oliveira conjecture”, Mathematics, 13:5 (2025), 711, 9 pp.
Supplementary files
