Abstract
The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.
References
[1] A. Brauer, Limits for the characteristic roots of a matrix IV. Applications to stochastic matrices. Duke Math. J. 19 (1952) 75-91.Search in Google Scholar
[2] M. Collao, C. R. Johnson, R. L. Soto, Universal realizability of spectra with two positive eigenvalues, Linear Algebra Appl. 545 (2018) 226-239.10.1016/j.laa.2018.01.039Search in Google Scholar
[3] J. D’Errico, Partitions of an integer, MathWorks®https://la.mathworks.com/matlabcentral/fileexchange/12009-partitions-of-an-integer (2018).Search in Google Scholar
[4] M. Fiedler, Eigenvalues of nonnegative symmetric matrices, Linear Algebra Appl. 9 (1974) 119-142.10.1016/0024-3795(74)90031-7Search in Google Scholar
[5] A.I. Julio, R.L. Soto, Persymmetric and bisymmetric nonnegative inverse eigenvalue problem, Linear Algebra Appl. 469 (2015) 130-152.10.1016/j.laa.2014.11.025Search in Google Scholar
[6] T. J. Laffey, M. E Meehan, A characterization of trace zero nonnegative 5 × 5 matrices, Linear Algebra Appl. 302-303 (1999) 295-302.10.1016/S0024-3795(99)00099-3Search in Google Scholar
[7] T.J. Laffey, H. Šmigoc, Nonnegative realization of spectra having negative real parts, Linear Algebra Appl. 416 (2006) 148-159.10.1016/j.laa.2005.12.008Search in Google Scholar
[8] R. Loewy, D. London, A note on the inverse problem for nonnegative matrices,Linear Multilinear Algebra 6 (1978) 83-90.10.1080/03081087808817226Search in Google Scholar
[9] D.S. Mackey, N. Mackey, S. Petrovic, Is every matrix similar to a Toeplitz matrix?, Linear Algebra Appl. 297 (1999) 87-105.10.1016/S0024-3795(99)00131-7Search in Google Scholar
[10] M. E. Meehan, Some results on matrix spectra, Ph. D. Thesis, National University of Ireland, Dublin, 1998.Search in Google Scholar
[11] M. Merca, A note on the determinant of a Toeplitz-Hessenberg matrix, Spec. Matrices 1 (2013) 10-16Search in Google Scholar
[12] H. Minc, Nonnegative matrices, John Wiley & sons, New York, 1988.Search in Google Scholar
[13] H. Perfect, Methods of constructing certain stochastic matrices II, Duke Math. J. 22 (1955) 305-311.10.1215/S0012-7094-55-02232-8Search in Google Scholar
[14] H. Šmigoc, The inverse eigenvalue problem for nonnegative matrices, Linear Algebra Appl. 393 (2004) 365-374.10.1016/j.laa.2004.03.036Search in Google Scholar
[15] R. L. Soto, Existence and construction of nonnegative matrices with prescribed spectrum, Linear Algebra Appl. 369 (2003) 169-184.10.1016/S0024-3795(02)00731-0Search in Google Scholar
[16] R.L. Soto, O. Rojo, Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem, Linear Algebra Appl. 416 (2006) 844-856.10.1016/j.laa.2005.12.026Search in Google Scholar
[17] R.L. Soto, O. Rojo, J. Moro, A. Borobia, Symmetric nonnegative realization of spectra, Electron. J. Linear Algebra 16 (2007) 1-18.10.13001/1081-3810.1178Search in Google Scholar
[18] R.L. Soto, R.C. Diaz, H. Nina, M. Salas, Nonnegative matrices with prescribed spectrum and elementary divisors, Linear Algebra Appl. 439 (2013) 3591-3604.10.1016/j.laa.2013.09.034Search in Google Scholar
[19] R.L. Soto, A.I. Julio, M. Salas, Nonnegative persymmetric matrices with prescribed elementary divisors, Linear Algebra Appl. 483 (2015) 130-157.10.1016/j.laa.2015.05.032Search in Google Scholar
[20] H.R. Suleimanova, Stochastic matrices with real characteristic values, Dokl. Akad. Nauk SSSR 66 (1949) 343-345.Search in Google Scholar
[21] J. Torre-Mayo, M. R. Abril-Raimundo, E. Alarcia-Estévez, C. Marijuán, M. Pisonero, The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs, Linear Algebra Appl. 426 (2007) 729-773.10.1016/j.laa.2007.06.014Search in Google Scholar
© 2019 Macarena Collao et al., published by De Gruyter
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