Abstract
In this article, we consider the problems (unsolved in the literature) of computing the nearest normal matrix X to a given non-normal matrix A, under certain constraints, that are (i) if A is real, we impose that also X is real; (ii) if A has known entries on a given sparsity pattern Ω and unknown/uncertain entries otherwise, we impose to X the constraint xij = aij for all entries (i,j) in the pattern Ω. As far as we know, there do not exist in the literature specific algorithms aiming to solve these problems. For the case in which all entries of A can be modified, there exists an algorithm by Ruhe, which is able to compute the closest normal matrix. However, if A is real, the closest computed matrix by Ruhe’s algorithm might be complex, which motivates the development of a different algorithm preserving reality. Normality is characterized in a very large number of ways; in this article, we consider the property that the square of the Frobenius norm of a normal matrix is equal to the sum of the squares of the moduli of its eigenvalues. This characterization allows us to formulate as equivalent problem the minimization of a functional of an unknown matrix, which should be normal, fulfill the required constraints, and have minimal distance from the given matrix A.
Similar content being viewed by others
References
Bakonyi, M., Woerdeman, H.J.: Matrix Completion, moments, and sums of hermitian squares. Princeton University Press (2011)
Bebiano, N., Furtado, S.: Structured distance to normality of tridiagonal matrices. Linear Algebra Appl. 552, 239–255 (2018)
Barl, F.: Higher-Rank Numerical Range of Almost Normal Matrices. Master’s Thesis, Univ. Krakow (2014)
Bathia, R., Choi, M.D.: Corners of normal matrices. Proc. Indian Acad. Sci. Math Sci. 116, 393–399 (2006)
Chu, M.T.: Least squares approximation by real normal matrices with specified spectrum. SIAM J. Matrix Anal. Appl. 12, 115–127 (1991)
Elsner, L., Ikramov, K.D.: Normal matrices: an update. Linear Algebra Appl. 285, 291–303 (1998)
Elsner, L., Paardekooper, M.H.C.: On measures of nonnormality of matrices. Linear Algebra Appl. 92, 107–124 (1987)
Gabriel, R.: The normal Δ h-matrices with connection to some Jacobi-like methods. Linear Algebra and Appl 91, 181–194 (1987)
Goldstine, H., Horwitz, L. P.: A procedure for the diagonalization of normal matrices. JACM 6, 176–195 (1959)
Grone, R., Johnson, C.R., Sa, E.R., Wolkowicz, H.: Normal matrices. Linear Algebra Appl. 92, 213–225 (1987)
Guglielmi, N., Lubich, C.: Differential equations for roaming pseudospectra: paths to extremal points and boundary tracking. SIAM J. Matrix Anal. Appl. 49, 1194–1209 (2011)
Guglielmi, N., Lubich, C.: Low-rank dynamics for computing extremal points of real pseudospectra. SIAM J. Matrix Anal. Appl. 34, 40–66 (2013)
Guglielmi, N., Mehrmann, V., Lubich, C.: On the nearest singular matrix pencil. SIAM J. Matrix Anal. Appl. 38, 776–806 (2017)
Henrici, P.: Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices. Numer. Math. 4, 24–40 (1962)
Higham, N.J.: Normal matrices: an update. Proceedings of the IMA Conference on Applications of Matrix Theory, pp. 1–27. Oxford University Press (1988)
Hall, M., Ryser, H.J.: Normal completions of incidence matrices. Am. J. Math. 76, 581–589 (1954)
Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1995)
Kaliuzhnyi-Verbovetskyi, D., Spitkovsky, I., Woerdeman, H.J.: Matrices with normal defect one. Oper. Matrices 30, 401–438 (2009)
Lee, S.L.: A practical upper bound for departure from normality. SIAM J. Matrix Anal. Appl. 16, 462–468 (1995)
Lee, S.L.: Best available bounds for departure from normality. SIAM J. Matrix Anal. Appl. 17, 984–991 (1996)
Li, C.K., Sze, N.S.: Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equation. Proc. Amer. Math. Soc. 136, 3013–3023 (2008)
Noschese, S., Pasquini, L., Reichel, L.: The structured distance to normality of an irreducible real tridiagonal matrix. Electronic. Trans. Numer. Anal. 23, 65–77 (2007)
Noschese, S., Reichel, L.: The structured distance to normality of Toeplitz matrices with application to preconditioning. Linear Algebra Appl. 18, 429–447 (2007)
Noschese, S., Reichel, L.: The structured distance to normality of banded Toeplitz matrices. BIT Numer. Math. 49, 629–640 (2009)
Noschese, S., Reichel, L.: The structured distance to normality of Toeplitz matrices with application to preconditioning. Numer. Linear Algebra Appl. 18, 429–447 (2011)
Psarrakos, P., Tsatsomeros, M.: Numerical range: (in) a matrix nutshell (2012)
Ruhe, A.: Closest normal matrix finally found!. BIT 27, 585–598 (1987)
Trefethen, L.N., Embree, R.: Spectra and pseudospectra : The behaviour of nonnormal matrices and operators. Princeton University Press (1993)
Woerdeman, H.J.: The separability problem and normal completions. Linear Algebra Appl. 376, 85–95 (2004)
Acknowledgments
The authors thank the anonymous referees for their helpful comments and suggestions. N. Guglielmi also thanks the Italian M.I.U.R. (Ministero dell’Istruzione dell’Universita’ e della Ricerca).
Funding
The authors thank the Italian INdAM-GNCS (Gruppo Nazionale di Calcolo Scientifico) for financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Lothar Reichel
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Guglielmi, N., Scalone, C. Computing the closest real normal matrix and normal completion. Adv Comput Math 45, 2867–2891 (2019). https://doi.org/10.1007/s10444-019-09717-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-019-09717-6
Keywords
- Normal completion
- Closest real normal matrix
- Closest normal matrix
- Normal completion of minimal norm
- Matrix nearness problems
- Matrix ODEs