Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 83–101 | Cite as

Positive semidefinite univariate matrix polynomials

  • Christoph HanselkaEmail author
  • Rainer Sinn


We study sum-of-squares representations of symmetric univariate real matrix polynomials that are positive semidefinite along the real line. We give a new proof of the fact that every positive semidefinite univariate matrix polynomial of size \(n\times n\) can be written as a sum of squares \(M=Q^TQ\), where Q has size \((n+1)\times n\), which was recently proved by Blekherman–Plaumann–Sinn–Vinzant. Our new approach using the theory of quadratic forms allows us to prove the conjecture made by these authors that these minimal representations \(M=Q^TQ\) are generically in one-to-one correspondence with the representations of the nonnegative univariate polynomial \(\det (M)\) as sums of two squares. In parallel, we will use our methods to prove the more elementary hermitian analogue that every hermitian univariate matrix polynomial M that is positive semidefinite along the real line, is a square, which is known as the matrix Fejér–Riesz Theorem.


Matrix factorizations Matrix polynomial Sum of squares Smith normal form 

Mathematics Subject Classification

Primary 14P05 Secondary 47A68 11E08 11E25 13J30 



We are grateful to Markus Schweighofer. Our approach extends fruitful discussions with him. The first author is supported by the Faculty Research Development Fund (FRDF) of The University of Auckland (Project no. 3709120). The second author would like to thank Bernd Sturmfels and the Max-Planck-Institute in Leipzig for their hospitality and support.


  1. 1.
    Blekherman, G., Plaumann, D., Sinn, R., Vinzant, C.: Low-rank sum-of-squares representations on varieties of minimal degree. Int. Math. Res. Not. (2016).
  2. 2.
    Cassels, J.W.S.: On the representation of rational functions as sums of squares. Acta Arith. 9, 79–82 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Choi, M.D., Lam, T.Y., Reznick, B.: Real zeros of positive semidefinite forms. I. Math. Z. 171(1), 1–26 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dritschel, M.A., Rovnyak, J.: The operator Fejér-Riesz theorem. In: Axler, S., Rosenthal, P., Sarason, D. (eds.) A Glimpse at Hilbert Space Operators: Paul R. Halmos in Memoriam, pp. 223–254. Springer, Basel (2010).
  5. 5.
    Eisenbud, D.: Commutative Algebra. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995) (with a view toward algebraic geometry) Google Scholar
  6. 6.
    Ephremidze, L., Selesnick, I., Spitkovsky, I.: On non-optimal spectral factorizations. Georgian Math. J. 24(4), 517–522 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fernando, J.F., Ruiz, J.M., Scheiderer, C.: Sums of squares of linear forms. Math. Res. Lett. 13(5–6), 947–956 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lam, T.Y.: Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics, vol. 67. American Mathematical Society, Providence (2005)zbMATHGoogle Scholar
  9. 9.
    Leep, D.B.: Sums of squares of polynomials and the invariant \(g_n(r)\). Unpublished manuscript, personal communication (2006)Google Scholar
  10. 10.
    MacDuffee, C.C.: The theory of matrices. Zentralblatt MATH 59(02), 347–462 (1933)Google Scholar
  11. 11.
    Neukirch, J.: Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322. Springer, Berlin (1999)zbMATHGoogle Scholar
  12. 12.
    O’Meara, O.T.: Introduction to Quadratic Forms. Springer, New York (1971) (second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 117) Google Scholar
  13. 13.
    Prestel, A., Delzell, C.N.: Positive Polynomials. Springer Monographs in Mathematics. Springer, Berlin (2001) (from Hilbert’s 17th problem to real algebra) Google Scholar
  14. 14.
    Pfister, A.: Multiplikative quadratische Formen. Arch. Math. (Basel) 16, 363–370 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tignol, J.-P.: A Cassels–Pfister theorem for involutions on central simple algebras. J. Algebra 181(3), 857–875 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wiener, N., Masani, P.: The prediction theory of multivariate stochastic processes. I. The regularity condition. Acta Math. 98, 111–150 (1957)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand
  2. 2.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany

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