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Algebra  2014

The Matrix Equation over Fields or Rings

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Abstract:

Let and let be an algebraically closed field with characteristic 0 or greater than . We show that if and satisfy , then are simultaneously triangularizable. Let be a reduced ring such that is not a zero divisor and let be a generic matrix over ; we show that is the sole solution of . Let be a commutative ring with unity; let be similar to such that, for every is not a zero divisor. If is a nilpotent solution of where , then . 1. Introduction Let be an integer at least . In the first part, we consider matrices with entries in , a field such that its characteristic is or greater than , and denotes its algebraic closure. In the second part, the entries of the matrices are elements of , a commutative ring with unity ; then, is the ring of endomorphisms of the free -module . Let and let be a polynomial with coefficients in the complex field such that . In [1, 2], the matrix is given and the authors study the matrix equations in the unknown : In the present paper, we extend some results obtained in [1, 2] when we replace with or . Two matrices , are said to be simultaneously triangularizable (abbreviated to ) over if there exists such that and are upper triangular matrices. The following result is a slight improvement of [1, Theorem 1] or of [3, Theorem 11′]. Theorem 1. One assumes that or is and that is a polynomial with coefficients in : Then, , are over . Remark 2. Note that the previous result fails when . We now consider (1) and (2). Let be a commutative ring with unity and let be a matrix where the are commuting indeterminates. Let be the ring of the polynomials in the indeterminates and with coefficients in . Thus, the algebra generated by is included in . In particular, there are no polynomial relations, with coefficients in , linking the . We say that is a generic matrix over . When is reduced (for every , implies ), we obtain a precise result. Proposition 3. Let , and let be a reduced ring such that is not a zero divisor. Let be a generic matrix. Then, is the sole solution of (1). Otherwise, we only obtain a partial result. Proposition 4. Let be a commutative ring with unity, , and let be a polynomial with coefficients in such that . Let be a nilpotent solution of (2). Then, all elements of the two-sided ideal, in , generated by are nilpotent. Yet, if is diagonalizable and its spectrum is “good,” then we obtain a complete solution. Theorem 5. Let be similar over to with such that, for every , is not a zero divisor. Let be a nilpotent solution of (2). (i)Then, there is such that where, for every , and .(ii)If moreover is a unit, then . 2. Equation

References

 [1] G. Bourgeois, “How to solve the matrix equation ,” Linear Algebra and Its Applications, vol. 434, no. 3, pp. 657–668, 2011. [2] D. Burde, “On the matrix equation ,” Linear Algebra and its Applications, vol. 404, pp. 147–165, 2005. [3] H. Shapiro, “Commutators which commute with one factor,” Pacific Journal of Mathematics, vol. 181, no. 3, pp. 323–336, 1997. [4] R. Borcherds, Berkeley University, mathematics 261AB, lecture 11, http://math.berkeley.edu/~reb/courses/261/11.pdf. [5] J. P. Anker and B. Orsted, Eds., Lie Theory, vol. 228 of Progress in Mathematics, Birkh？user, 2004. [6] V. Baranovsky, “The variety of pairs of commuting nilpotent matrices is irreducible,” Transformation Groups, vol. 6, no. 1, pp. 3–8, 2001. [7] A. Bostan and T. Combot, “A binomial-like matrix equation,” The American Mathematical Monthly, vol. 119, no. 7, pp. 593–597, 2012. [8] T. S. Motzkin and O. Taussky, “Pairs of matrices with property II,” Transactions of the American Mathematical Society, vol. 80, pp. 387–401, 1955. [9] P. M. Cohn, Basic Algebra: Groups, Rings and Fields, Springer, London, UK, 2003. [10] J. Bra？i？, “On the Jacobson's lemma,” http://jankobracic.files.wordpress.com/2011/02/on-the-jacobsons-lemma.pdf. [11] J. Bra？i？ and B. Kuzma, “Localizations of the Kleinecke-Shikorov theoerem,” Operators and Matrices, vol. 1, no. 3, pp. 385–389, 2007. [12] G. Almkvist, “Endomorphisms of finetely generated projective modules over a commutative ring,” Arkiv f？r Matematik, vol. 11, no. 1-2, pp. 263–301, 1973. [13] N. H. McCoy, “On the characteristic roots of matric polynomials,” Bulletin of the American Mathematical Society, vol. 42, no. 8, pp. 592–600, 1936. [14] N. H. McCoy, “A theorem on matrices over a commutative ring,” Bulletin of the American Mathematical Society, vol. 45, pp. 740–744, 1939. [15] W. Brown, Matrices over Commutative Rings, Taylor & Francis, Boca Raton, Fla, USA, 1992.

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