Perron-Frobenius theory on the higher-rank numerical range for some classes of real matrices

Document Type : Research Paper


1 Department of Mathematics, University of Hormozgan, Bandar Abbas, Iran

2 Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.


We present an extension of Perron-Frobenius theory to the higher-rank numerical range
of real matrices. We define a new type of the rank-k numerical radius for real matrices, i.e., the
sign-real rank-k numerical radius‎, and derive some properties of it. In addition, we extend Issos' results on the higher-rank numerical range of nonnegative matrices to real matrices.‎
Finally, we give some examples that are used to illustrate our theoretical results.‎‎


[1] A. Aretaki and J. Maroulas, The higher rank numerical range of nonnegative matrices,Cent. Eur. J. Math., vol. 11, no.3(2013), 435{446.
[2] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,vol. 9, SIAM, Philadelphia, 1994.
[3] M.T. Chien and H. Nakazato, Boundary generating curves of the c-numerical range,Linear Algebra Appl., vol. 294, no. 1-3(1999), 67{84.
[4] M.D. Choi, D.W. Kribs and K. Zyczkowski, Higher-rank numerical ranges and compression problems, Linear Algebra Appl., vol. 418 no. 2-3(2006), 828{839.
[5] M.D. Choi, D.W. Kribs and K. Zyczkowski, Quantum error correcting codes from the compression formalism, Rep. Math. Phys., vol. 58, no. 1(2006), 77{86.
[6] G. Frobenius, Uber Matrizen aus nichtnegativen Elementen, Math. Nat. K1., (1912),456{477.
[7] R.A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
[8] R.A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press,Cambridge, 1991.
[9] J.N. Issos, The  eld of values of non-negative irreducible matrices, Ph.D. Thesis,Auburn University, 1966.
[10] C.K. Li, B.-S. Tam and P.Y. Wu, The numerical range of a nonnegative matrix, Linear Algebra Appl., vol. 350, no. 1-3(2002), 1{23.
[11] C.K. Li and H. Schneider, Applications of the Perron{Frobenius theory to population dynamics, J. Math. Biol., vol. 44, no. 5(2002), 250{262.
[12] C.K. Li, Y.T. Poon and N.S. Sze, Condition for the higher rank numerical range to be non-empty, Linear Multilinear Algebra, vol. 57, no. 4(2009), 365{368.
[13] M.-H. Matcovschi and O. Pastravanu, Perron{Frobenius theorem and invariant sets in linear systems dynamics , in Proceedings of the 15th IEEE Mediterranean Conference on Control and Automation (MED07), Athens, Greece, 2007.
[14] H. Mink, Nonnegative Matrices, Wiley, New York, 1988.
[15] S.M. Rump., Theorems of Perron-Frobenius type for matrices without sign restrictions,linear Algebra Appl., vol. 266(1997), 1{42.
[16] S. M. Rump, Conservatism of the Circle Criterion Solution of a Problem posed by A.Megretski, IEEE Trans. Autom. Control, vol. 46, no. 10(2001), 1605{1608.
[17] S. M. Rump, conditioned Matrices are componentwise near to singularity, SIAM Rev.,vol. 41, no. 1(1999), 102{112.
[18] B. Shafai, J. Chen and M. Kothandaraman, Explicit formulas for stability of nonnegative and Metzlerian matrices, IEEE Transactions on Automatic Control, vol. 42, no. 2(1997),265{270.
[19] M. Zangiabadi and H. R. Afshin, A new concept for numerical radius: the sign-real numerical radius, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., vol.76, no. 3(2014), 91{98.
[20] M. Zangiabadi and H. R. Afshin, Perron{Frobenius theory on the numerical range for some classes of real matrices, J. Mahani Math. Res. Cent., vol. 2, no. 2(2013), 1{15.
Volume 10, Issue 2
Special Issue Dedicated to Professor M. Radjabalipour on the occasion of his 75th birthday.
October 2021
Pages 49-61
  • Receive Date: 11 May 2021
  • Revise Date: 29 July 2021
  • Accept Date: 05 August 2021