Abstract
There exists a large set of real symmetric matrices whose entries are linear functions in several variables such that each matrix in this set is definite at some point, that is, the matrix is definite after substituting some numbers for variables. In particular, this property holds for almost all such matrices of order two with entries depending on two variables. The same property holds for almost all matrices of order two with entries depending on a larger number of variables when this number exceeds the order of the matrix. Some examples are discussed in detail. Some asymmetric matrices are also considered. In particular, for almost every matrix whose entries are linear functions in several variables, the determinant of the matrix is positive at some point and negative at another point.
Similar content being viewed by others
REFERENCES
V. I. Zabotin and Yu. A. Chernyaev, “Newton’s method for minimizing a convex twice differentiable function on a preconvex set,” Comput. Math. Math. Phys. 58 (3), 322–327 (2018). https://doi.org/10.1134/S0965542518030144
N. N. Vorob’ev, Jr., and D. Yu. Grigor’ev, “Finding connected components of a semialgebraic set in subexponential time,” J. Math. Sci. 70 (4), 1847–1872 (1994). https://doi.org/10.1007/BF02112426
Yu. G. Evtushenko, M. A. Posypkin, L. A. Rybak, and A. V. Turkin, “Finding sets of solutions to systems of nonlinear inequalities,” Comput. Math. Math. Phys. 57 (8), 1241–1247 (2017). https://doi.org/10.1134/S0965542517080073
M. Safey El Din and E. Schost, “A nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets,” J. ACM 63 (6), Article No. 48 (2017).https://doi.org/10.1145/2996450
C. J. Hillar and L.-H. Lim, “Most tensor problems are NP-hard,” J. ACM 60 (6), Article No. 45 (2013).https://doi.org/10.1145/2512329
N. A. Sal’kov, “General principles for formation of ruled surfaces. Part 1,” Geom. Grafika 6 (4), 20–31 (2018). https://doi.org/10.12737/article_5c21f4a06dbb74.56415078
V. G. Zhadan, “Primal Newton method for the linear cone programming problem,” Comput. Math. Math. Phys. 58 (2), 207–214 (2018). https://doi.org/10.1134/S0965542518020173
J. Renegar, “Accelerated first-order methods for hyperbolic programming,” Math. Program. Ser. A 173, 1–35 (2019).https://doi.org/10.1007/s10107-017-1203-y
B. F. Lourenco, T. Kitahara, M. Muramatsu, and T. Tsuchiya, “An extension of Chubanov’s algorithm to symmetric cones,” Math. Program. Ser. A 173, 117–149 (2019).https://doi.org/10.1007/s10107-017-1207-7
A. Y. Aravkin, J. V. Burke, D. Drusvyatskiy, M. P. Friedlander, and S. Roy, “Level-set methods for convex optimization,” Math. Program. Ser. B 174, 359–390 (2019).https://doi.org/10.1007/s10107-018-1351-8
M. Laurent and A. Varvitsiotis, “Positive semidefinite matrix completion, universal rigidity, and the strong Arnold property,” Linear Algebra Appl. 452, 292–317 (2014).https://doi.org/10.1016/j.laa.2014.03.015
A. Kurpisz, S. Leppanen, and M. Mastrolilli, “Sum-of-squares hierarchy lower bounds for symmetric formulations,” Math. Program. Ser. A (2019). https://doi.org/10.1007/s10107-019-01398-9.
G. Braun, S. Pokutta, and D. Zink, “Affine reductions for LPs and SDPs,” Math. Program. Ser. A 173, 281–312 (2019).https://doi.org/10.1007/s10107-017-1221-9
Z. Huang and X. Zhan, “Nonsymmetric normal entry patterns with the maximum number of distinct indeterminates,” Linear Algebra Appl. 485, 359–371 (2015).https://doi.org/10.1016/j.laa.2015.08.003
H. H. Van and R. Quinlan, “On the maximum rank of completions of entry pattern matrices,” Linear Algebra Appl. 525, 1–19 (2017).https://doi.org/10.1016/j.laa.2017.02.035
H. H. Van and R. Quinlan, “Almost-nonsingular entry pattern matrices,” Linear Algebra Appl. 578, 334–355 (2019).https://doi.org/10.1016/j.laa.2019.05.006
R. A. Brualdi, Z. Huang, and X. Zhan, “Singular, nonsingular, and bounded rank completions of ACI-matrices,” Linear Algebra Appl. 433, 1452–1462 (2010).https://doi.org/10.1016/j.laa.2010.05.018
J. McTigue and R. Quinlan, “Partial matrices of constant rank,” Linear Algebra Appl. 446, 177–191 (2014).https://doi.org/10.1016/j.laa.2013.12.020
A. Borobia and R. Canogar, “ACI-matrices of constant rank over arbitrary fields,” Linear Algebra Appl. 527, 232–259 (2017).https://doi.org/10.1016/j.laa.2017.04.002
A. Yu. Nikitin and A. N. Rybalov, “On complexity of the satisfiability problem of systems over finite posets,” Prikl. Diskret. Mat., No. 39, 94–98 (2018). https://doi.org/10.17223/20710410/39/8
A. N. Rybalov, “Generic amplification of recursively enumerable sets,” Algebra Logic 57 (4), 289–294 (2018). https://doi.org/10.1007/s10469-018-9500-y
A. N. Rybalov, “On generic complexity of decidability problem for Diophantine systems in the Skolem’s form,” Prikl. Diskret. Mat., No. 37, 100–106 (2017). https://doi.org/10.17223/20710410/37/8
G. I. Malashonok, “MathPartner computer algebra,” Program. Comput. Software 43 (2), 112–118 (2017). https://doi.org/10.1134/S0361768817020086
A. V. Smirnov, “The bilinear complexity and practical algorithms for matrix multiplication,” Comput. Math. Math. Phys. 53 (12), 1781–1795 (2013). https://doi.org/10.1134/S0965542513120129
A. V. Smirnov, “A bilinear algorithm of length 22 for approximate multiplication of 2 × 7 and 7 × 2 matrices,” Comput. Math. Math. Phys. 55 (4), 541–545 (2015). https://doi.org/10.1134/S0965542515040168
V. Ya. Pan, “Fast matrix multiplication and its algebraic neighborhood,” Sb. Math. 208 (11), 1661–1704 (2017). https://doi.org/10.1070/SM8833
M. Cenk and M. A. Hasan, “On the arithmetic complexity of Strassen-like matrix multiplications,” J. Symbolic Comput. 80 (2), 484–501 (2017).https://doi.org/10.1016/j.jsc.2016.07.004
D. A. Dolgov, “Polynomial greatest common divisor as a solution of system of linear equations,” Lobachevskii J. Math. 39 (7), 985–991 (2018).https://doi.org/10.1134/S1995080218070090
A. Bernardi, G. Blekherman, and G. Ottaviani, “On real typical ranks,” Boll. Unione Mat. Ital. 11 (3), 293–307 (2018).https://doi.org/10.1007/s40574-017-0134-0
ACKNOWLEDGMENTS
The author is grateful to the reviewer for the remarks made.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
Seliverstov, A.V. Symmetric Matrices Whose Entries Are Linear Functions. Comput. Math. and Math. Phys. 60, 102–108 (2020). https://doi.org/10.1134/S0965542520010121
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542520010121