Abstract
We call a nonscalar matrix maximal (or minimal) if its centralizer is maximal (respectively minimal) in the poset of all centralizers of matrices. We discuss the form of maximal and minimal matrices in the algebra of upper triangular matrices.
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Aiat Hadj Ahmed, D., Słowik, R.: m-commuting maps of the rings of infinite triangular and strictly triangular matrices. (in preparation)
Arenas-Herrera, M.I., Verde-Star, L.: Representation of doubly infinite matrices as non-commutative Laurent series. Special Matrices 5, 250–257 (2017)
Bhatia, R., Rosenthal, P.: How and why to solve the operator equation \(AX-XB=Y\). Bull. Lond. Math. Soc. 29, 1–21 (1997)
Brown, W.C.: Constructing maximal commutative subalgebra of matrix rings in small dimensions. Commun. Algebra 25, 3923–3946 (1997)
Dolinar, G., Guterman, A., Kuzma, B., Oblak, P.: Commuting graphs and extremal centralizers. Ars Math. Contemp. 7, 453–459 (2014)
Dolinar, G., Guterman, A., Kuzma, B., Oblak, P.: Extremal matrix centralizers. Linear Algebra Appl. 438, 2904–2910 (2013)
Drazin, M.: Some generalizations of matrix commutativity. Proc. Lond. Math. Soc. 3–1, 222–231 (1951)
Drazin, M.P., Dungey, J.W., Gruenberg, K.W.: Some theorems on commutative matrices. J. Lond. Math. Soc. 26, 221–228 (1951)
Frobenius, G.: Über lineare Substitution und Bilinear Formen, Vol. 84 , pp. 1–63. Grelle (1878)
Gerstenhaber, M.: On dominance and varieties of commuting matrices. Ann. Math. 73, 324–348 (1961)
Guterman, A.E., Markova, O.V.: Commutative matrix subalgebras and length function. Linear Algebra Appl. 430, 1790–1805 (2009)
Hoffman, K., Kunze, R.: Linear Algebra, 2nd edn. Prentice-Hall In, Engelwood Cllifs (1971)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
Lagerstrom, P.: A proof of a theorem on commutative matrices. Bull. Am. Math. Soc. 51, 535–536 (1945)
Šemrl, P.: Non-linear commutativity preserving maps. Acta Sci. Math. (Szeged) 71, 781–819 (2005)
Słowik, R.: Maps on infinite triangular matrices preserving idempotents. Linear Multilinear Algebra 62, 938–964 (2014)
Słowik, R.: Every infinite triangular matrix is similar to a generalized infinite Jordan matrix. Linear Multilinear Algebra 65, 1362–1373 (2017)
Słowik, R.: Corrigendum to ‘Every infinite triangular matrix is similar to a generalized infinite Jordan matrix (Linear Multilinear Algebra 65, 1362–1373 (2017))’. Linear Multilinear Algebra. https://doi.org/10.1080/03081087.2017.1397094
Thijsse, P.: Upper triangular similarity of upper triangular similarity of upper triangular matrices. Linear Algebra Appl. 260, 119–149 (1997)
Wang, S.: The Jordan normal form of infinite matrices. Chin. Sci. Bull. 41, 1943–1946 (1996)
Wedderburn, J.H.M.: Lectures on Matrices. Dover Publications Inc, New York (1964)
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Słowik, R. Maximal and Minimal Triangular Matrices. Results Math 73, 58 (2018). https://doi.org/10.1007/s00025-018-0819-4
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DOI: https://doi.org/10.1007/s00025-018-0819-4