Guterman, A. E.; Markova, O. V. Commutative matrix subalgebras and length function. (English) Zbl 1160.13016 Linear Algebra Appl. 430, No. 7, 1790-1805 (2009). It is proved that the upper bound for the length of a commutative matrix subalgebra in \(M_n(\mathbb C)\) is equal to \(n-1\). In Sections 2-5 authors give some basic properties of the length function. In Section 8 it is demonstrated that there are maximal commutative subalgebras of nonmaximal length. Reviewer: Yueh-er Kuo (Knoxville) Cited in 2 ReviewsCited in 28 Documents MSC: 13E10 Commutative Artinian rings and modules, finite-dimensional algebras 15A27 Commutativity of matrices 15A18 Eigenvalues, singular values, and eigenvectors Keywords:finite-dimensional algebras; matrix length; commutative matrix algebras × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Brown, W. C.; Call, F. W., Maximal commutative subalgebras of \(n \times n\) matrices, Comm. Algebra, 21, 12, 4439-4460 (1993) · Zbl 0796.16021 [2] Courter, R. C., The dimension of maximal commutative subalgebras of \(K_n\), Duke Math. J., 32, 225-232 (1965) · Zbl 0141.03101 [3] Gerstenhaber, M., On dominance and varieties of commuting matrices, Ann. Math., 73, 2, 324-348 (1961) · Zbl 0168.28201 [4] Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press · Zbl 0576.15001 [5] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge University Press · Zbl 0729.15001 [6] Laffey, T. J., The minimal dimension of maximal commutative subalgebras of full matrix algebras, Linear Algebra Appl., 71, 199-212 (1985) · Zbl 0578.15008 [7] Laffey, T. J.; Lazarus, S., Two-generated commutative matrix subalgebras, Linear Algebra Appl., 147, 249-273 (1991) · Zbl 0718.15008 [8] Mal’cev, A. I., Basics of Linear Algebra (1975), Nauka: Nauka Moscow, (in Russian) [9] Markova, O. V., On the length of upper-triangular matrix algebra, Uspekhi Matem. Nauk, 60, 5, 177-178 (2005), (in Russian). English translation: Russian Mathematical Surveys 60 (5) (2005) 984-985 · Zbl 1140.16306 [10] Markova, O. V., Length computation of matrix subalgebras of special type, Fundam. Appl. Math., 13, 4, 165-197 (2007), (in Russian) [11] Pappacena, C. J., An upper bound for the length of a finite-dimensional algebra, J. Algebra, 197, 535-545 (1997) · Zbl 0888.16008 [12] Paz, A., An application of the Cayley-Hamilton theorem to matrix polynomials in several variables, Linear and Multilinear Algebra, 15, 161-170 (1984) · Zbl 0536.15007 [13] Pierce, R., Associative Algebras (1982), Springer-Verlag: Springer-Verlag Berlin · Zbl 0497.16001 [14] Schur, I., Zur Theorie der Vertauschbären Matrizen, J. Reine Angew. Math., 130, 66-76 (1905) · JFM 36.0140.01 [15] Spencer, A. J.M.; Rivlin, R. S., The theory of matrix polynomials and its applications to the mechanics of isotropic continua, Arch. Rat. Mech. Anal., 2, 309-336 (1959) · Zbl 0095.25101 [16] Spencer, A. J.M.; Rivlin, R. S., Further results in the theory of matrix polynomials, Arch. Rat. Mech. Anal., 4, 214-230 (1960) · Zbl 0095.25103 [17] Suprunenko, D. A.; Tyshkevich, R. I., Commutative Matrices (1968), Academic Press: Academic Press New York, NY · Zbl 1103.16302 [18] Wadsworth, A., The algebra generated by two commuting matrices, Linear and Multilinear Algebra, 27, 159-162 (1990) · Zbl 0703.15016 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.