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Corrigendum to “Spectra universally realizable by doubly stochastic matrices”

  • Macarena Collao , Mario Salas and Ricardo L. Soto EMAIL logo
Published/Copyright: January 2, 2023
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Special Matrices
From the journal Special Matrices Volume 11 Issue 1

Abstract

We correct an error in the statement and the proof of Theorem 2.1 and Corollary 2.1 in our previous study [Spec. Matrices 6 (2018), 301–309], Section 2: On nonnegative matrices similar to positive matrices.

Keywords: elementary divisor; nonnegative matrix; nonnegative inverse elementary divisors problem
MSC 2010: 15A18; 15A51

In a previous study [Spec. Matrices 6 (2018), 301–309], Section 2, Theorem 2.1 and Corollary 2.1, there is an error in assuming that if the matrix A is nonnegative with a positive row, then A T , which has a positive column, is similar to a positive matrix. The reason is that we cannot guarantee that A T has a positive eigenvector (although in many cases it does). Moreover, since it has been proved in [1,2] that if A is nonnegative irreducible with a positive row or column, then A is similar to a positive matrix, we do not consider the case irreducible.

Then, Theorem 2.1 and Corollary 2.1 in [3] are modified to:

Theorem 0.1

Let A ∈ C S λ 1 be a nonnegative matrix with spectrum Λ = { λ 1 , … , λ n } , λ 1 > ∣ λ i ∣ , i = 2 , … , n , and a positive column. Then A is similar to a positive matrix. If A is nonnegative with spectrum Λ and a positive row, and A T has a positive Perron eigenvector, then A is also similar to a positive matrix.

Proof

If A ∈ C S λ 1 is nonnegative with a positive column, say the last one, then A has a positive Perron eigenvector e T = [ 1 , 1 , … , 1 ] and for q T = [ q 1 , q 2 , … , q n ] , with

q i > 0 , i = 1 , 2 , … , n − 1 , ∑ i = 1 n q i = 0 , ∑ i = 1 n − 1 q i < min 1 ≤ i ≤ n a i n ,

A + eq T is positive with spectrum Λ . Thus, from Lemma 1.1, A is similar to A + eq T . If A is nonnegative with a positive row, and A T has a positive eigenvector x T = [ x 1 , … , x n ] , then for D = diag { x 1 , … , x n } ,

D − 1 A D = x j x i a i j ∈ C S λ 1

is nonnegative with a positive column and therefore it is similar to a positive matrix.□

Corollary 0.1

Let A be the matrix of Theorem 0.1. If A is diagonalizable nonnegative with a positive column, then A is similar to a diagonalizable positive matrix. If A is diagonalizable nonnegative with a positive row, and A T has a positive Perron eigenvector, then A is also similar to a diagonalizable positive matrix.

Proof

It is immediate from the aforementioned Theorem 0.1 and Lemma 1.1 in [3].□

  1. Funding information: The authors state no funding involved.

  2. Conflict of Interest: The authors state no conflict of interest.

  3. Data availability statement: All data generated or analyzed during this study are included in this published article [and its supplementary information files].

References

[1] A. Borobia and J. Moro, On nonnegative matrices similar to positive matrices, Linear Algebra Appl. 266 (1997), 365–379. 10.1016/S0024-3795(97)00362-5Search in Google Scholar

[2] T. J. Laffey, R. Loewy, and H. Šmigoc, Nonnegative matrices that are similar to positive matrices, SIAM J. Matrix Analysis Appl. 31 (2009), 629–649. 10.1137/070692807Search in Google Scholar

[3] M. Collao, M. Salas, and R. L. Soto, Spectra universally realizable by doubly stochastic matrices, Spec. Matrices 6 (2018), 301–309. 10.1515/spma-2018-0025Search in Google Scholar
Received: 2022-11-10
Revised: 2022-11-11
Accepted: 2022-11-11
Published Online: 2023-01-02

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/spma-2022-0179
Keywords for this article
elementary divisor; nonnegative matrix; nonnegative inverse elementary divisors problem
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Determinants of some Hessenberg matrices with generating functions
  3. On monotone Markov chains and properties of monotone matrix roots
  4. On the spectral properties of real antitridiagonal Hankel matrices
  5. The complete positivity of symmetric tridiagonal and pentadiagonal matrices
  6. Two n × n G-classes of matrices having finite intersection
  7. On new universal realizability criteria
  8. On inverse sum indeg energy of graphs
  9. Incidence matrices and line graphs of mixed graphs
  10. Diagonal dominance and invertibility of matrices
  11. New versions of refinements and reverses of Young-type inequalities with the Kantorovich constant
  12. W-MPD–N-DMP-solutions of constrained quaternion matrix equations
  13. Representing the Stirling polynomials σn(x) in dependence of n and an application to polynomial zero identities
  14. The effect of removing a 2-downer edge or a cut 2-downer edge triangle for an eigenvalue
  15. Idempotent operator and its applications in Schur complements on Hilbert C*-module
  16. On the distance energy of k-uniform hypergraphs
  17. The bipartite Laplacian matrix of a nonsingular tree
  18. Combined matrix of diagonally equipotent matrices
  19. Walks and eigenvalues of signed graphs
  20. On 3-by-3 row stochastic matrices
  21. Legendre pairs of lengths ℓ ≡ 0 (mod 5)
  22. Integral Laplacian graphs with a unique repeated Laplacian eigenvalue, I
  23. Communication
  24. Class of finite-dimensional matrices with diagonals that majorize their spectrum
  25. Corrigendum
  26. Corrigendum to “Spectra universally realizable by doubly stochastic matrices”
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