Abstract
The class of sparse companion matrices was recently characterized in terms of unit Hessenberg matrices. We determine which sparse companion matrices have the lowest bandwidth, that is, we characterize which sparse companion matrices are permutationally similar to a pentadiagonal matrix and describe how to find the permutation involved. In the process, we determine which of the Fiedler companion matrices are permutationally similar to a pentadiagonal matrix. We also describe how to find a Fiedler factorization, up to transpose, given only its corner entries.
References
[1] J.L. Aurentz, R. Vandebril, and D.S. Watkins, Fast computation of the zeros of a polynomial via factorization of the companion matrix, SIAM J. Sci. Comput.35 (2013) A255 – A269.10.1137/120865392Search in Google Scholar
[2] J.L. Aurentz, R. Vandebril, and D.S. Watkins, Fast computation of eigenvalues of companion, comrade, and related matrices, BIT Numer. Math.54 (2014) 7–30.10.1007/s10543-013-0449-xSearch in Google Scholar
[3] T. Bella, V. Olshevsky, and P. Zhlobich, A quasiseparable approach to five-diagonal CMV and Fiedler matrices, Linear Algebra Appl.434(4) (2011) 957–976.10.1016/j.laa.2010.10.011Search in Google Scholar
[4] B. Bevilacqua, G.M. Del Corso, and L. Gemignani, A CMV–based eigensolver for companion matrices, SIAM J. Matrix Anal. Appl.36(3) (2015) 1046–1068.10.1137/140978065Search in Google Scholar
[5] D.A. Bini, P. Boito, Y. Eidelman, L. Gemignani, and I. Gohberg, A fast implicit QR eigenvalue algorithm for companion matrices, Linear Algebra Appl.432(8) (2010) 2006–2031.10.1016/j.laa.2009.08.003Search in Google Scholar
[6] D.A. Bini, F. Daddi, and L. Gemignani, On the shifted QR iteration applied to companion matrices, Electron. Trans. Numer. Anal.18 (2004) 137–152.Search in Google Scholar
[7] S. Chandrasekaran, M. Gu, J. Xia, and J. Zhu, A fast QR algorithm for companion matrices, Oper. Theory Adv. Appl.179 (2008) 111–143.Search in Google Scholar
[8] F. De Terán, F. Dopico, and D.S. Mackey, Fiedler companion linearizations and the recovery of minimal indices, SIAM J. Matrix Anal. Appl., 31:4 (2010) 2181–2204.Search in Google Scholar
[9] F. De Terán, F. Dopico, and J. Pérez, Condition numbers for inversion of Fiedler companion matrices, Linear Algebra Appl.439 (2013) 944–981.10.1016/j.laa.2012.09.020Search in Google Scholar
[10] B. Eastman, I.-J. Kim, B. Shader, and K.N. Vander Meulen, Companion matrix patterns, Linear Algebra Appl.463 (2014) 255–272.10.1016/j.laa.2014.09.010Search in Google Scholar
[11] M. Fiedler, A note on companion matrices, Linear Algebra Appl.372 (2003) 325–331.10.1016/S0024-3795(03)00548-2Search in Google Scholar
[12] C. Garnett, B. Shader, C. Shader, and P. van den Driessche, Characterization of a family of generalized companion matrices Linear Algebra Appl. (2015), in press, http://dx.doi.org/10.1016/j.laa.2015.07.031.Search in Google Scholar
[13] N.J. Higham, D.S. Mackey, N. Mackey, and F. Tisseur, Symmetric linearizations for matrix polynomials, SIAM J. Matrix Anal. Appl.29 (2006) 143–159.10.1137/050646202Search in Google Scholar
[14] C. Ma and X. Zhan, Extremal sparsity of the companion matrix of a polynomial, Linear Algebra Appl.438 (2013) 621–625.10.1016/j.laa.2012.08.017Search in Google Scholar
[15] S. Vologiannidis and E.N. Antoniou, A permuted factors approach for the linearization of polynomial matrices, Math. Control Signals Systems22 (2011) 317–342.10.1007/s00498-011-0059-6Search in Google Scholar
© 2016 Brydon Eastman et al., published by De Gruyter Open
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