computational complexity

, Volume 26, Issue 3, pp 717–763 | Cite as

Non-commutative Edmonds’ problem and matrix semi-invariants

  • Gábor Ivanyos
  • Youming Qiao
  • K. V. Subrahmanyam


In 1967, J. Edmonds introduced the problem of computing the rank over the rational function field of an \({n \times n}\) matrix T with integral homogeneous linear polynomials. In this paper, we consider the non-commutative version of Edmonds’ problem: compute the rank of T over the free skew field. This problem has been proposed, sometimes in disguise, from several different perspectives in the study of, for example, the free skew field itself (Cohn in J Symbol Log 38(2):309–314, 1973), matrix spaces of low rank (Fortin-Reutenauer in Sémin Lothar Comb 52:B52f 2004), Edmonds’ original problem (Gurvits in J Comput Syst Sci 69(3):448–484, 2004), and more recently, non-commutative arithmetic circuits with divisions (Hrubeš and Wigderson in Theory Comput 11:357-393, 2015. doi: 10.4086/toc.2015.v011a014).

It is known that this problem relates to the following invariant ring, which we call the \({\mathbb{F}}\)-algebra of matrix semi-invariants, denoted as R(n, m). For a field \({\mathbb{F}}\), it is the ring of invariant polynomials for the action of \({{\rm SL}(n, \mathbb{F}) \times {\rm SL}(n, \mathbb{F})}\) on tuples of matrices—\({(A, C)\in {\rm SL}(n, \mathbb{F}) \times {\rm SL}(n, \mathbb{F})}\) sends \({(B_{1}, \ldots, B_m)\in M(n, \mathbb{F})^{\oplus m}}\) to \({(AB_1 {C}^{{\rm T}}, \ldots, AB_m {C}^{\rm T})}\). Then those T with non-commutative rank <  n correspond to those points in the nullcone of R(n, m). In particular, if the nullcone of R(n, m) is defined by elements of degree \({\leq \sigma}\), then there follows a \({{\rm poly}(n,\sigma)}\)-time randomized algorithm to decide whether the non-commutative rank of T is full. To our knowledge, previously the best bound for \({\sigma}\) was \({O(n^{2}\cdot 4^{n^2})}\) over algebraically closed fields of characteristic 0 (Derksen in Proc Am Math Soc 129(4):955–964, 2001).

We now state the main contributions of this paper:
  • We observe that by using an algorithm of Gurvits, and assuming the above bound \({\sigma}\) for R(n, m) over \({\mathbb{Q}}\), deciding whether or not T has non-commutative rank < n over \({\mathbb{Q}}\) can be done deterministically in time polynomial in the input size and \({\sigma}\).

  • When \({\mathbb{F}}\) is large enough, we devise an algorithm for the non-commutative Edmonds problem which runs in time polynomial in (n + 1)!. Furthermore, due to the structure of this algorithm, we also have the following results.
    • If the commutative rank and the non-commutative rank of T differ by a constant there exists a randomized efficient algorithm to compute the non-commutative rank of T. This improves upon a result of Fortin and Reutenauer, who gave a randomized efficient algorithm to decide whether the commutative and non-commutative ranks are equal.

    • We show that \({\sigma\leq (n+1)!}\). This not only improves the bound obtained from Derksen’s work over algebraically closed field of characteristic 0 but, more importantly, also provides for the first time an explicit bound on \({\sigma}\) for matrix semi-invariants over fields of positive characteristics. Furthermore, this does not require \({\mathbb{F}}\) to be algebraically closed.


Edmonds’ problem symbolic determinant identity test semi-invariants of quivers non-commutative rank 

Subject classification

13A50 68W30 


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  1. B. Adsul, S. Nayak & K. V. Subrahmanyam (2007). A geometric approach to the Kronecker problem II: rectangular shapes, invariants of matrices and the Artin–Procesi theorem. Preprint.Google Scholar
  2. Adsul Bharat, Subrahmanyam K.V. (2008) A geometric approach to the Kronecker problem I: the two row case. Proceedings Mathematical Sciences 118(2): 213–226MathSciNetCrossRefzbMATHGoogle Scholar
  3. S.A Amitsur (1966). Rational identities and applications to algebra and geometry. Journal of Algebra 3(3), 304 – 359. ISSN 0021-8693. URL
  4. MD Atkinson & S Lloyd (1981). Primitive spaces of matrices of bounded rank. Journal of the Australian Mathematical Society (Series A) 30(04), 473–482.Google Scholar
  5. George W. Bergman (1969–1970). Skew fields of noncommutative rational functions (preliminary version). Séminaire Schützenberger 1, 1–18. URL
  6. Peter A. Brooksbank, Eugene M. Luks (2008) Testing isomorphism of modules. Journal of Algebra 320(11): 4020–4029MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bürgin M., Draisma J. (2006) The Hilbert null-cone on tuples of matrices and bilinear forms. Mathematische Zeitschrift 254(4): 785–809MathSciNetCrossRefzbMATHGoogle Scholar
  8. Peter Bürgisser, J. M. Landsberg, Laurent Manivel & Jerzy Weyman (2011). An Overview of Mathematical Issues Arising in the Geometric Complexity Theory Approach to \({VP \neq VNP}\). SIAM J. Comput. 40(4), 1179–1209. doi: 10.1137/090765328.
  9. Buss Jonathan F., Frandsen Gudmund S., Shallit Jeffrey O. (1999) The computational complexity of some problems of linear algebra. J. Comput. Syst. Sci. 58(3): 572–596MathSciNetCrossRefzbMATHGoogle Scholar
  10. Marco Carmosino, Russell Impagliazzo, Valentine Kabanets & Antonina Kolokolova (2015). Tighter Connections between Derandomization and Circuit Lower Bounds. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2015, August 24–26, 2015, Princeton, NJ, USA, 645–658. doi: 10.4230/LIPIcs.APPROX-RANDOM.2015.645.
  11. Alexander L. Chistov, Gábor Ivanyos & Marek Karpinski (1997). Polynomial Time Algorithms for Modules over Finite Dimensional Algebras. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ISSAC ’97, Maui, Hawaii, USA, July 21–23, 1997, 68–74.Google Scholar
  12. Ajeh M Cohen, Gábor Ivanyos & David B Wales (1997). Finding the radical of an algebra of linear transformations. Journal of Pure and Applied Algebra 117, 177–193.Google Scholar
  13. P. M. Cohn (1973). The Word Problem for Free Fields. J. Symbolic Logic 38(2), 309–314. URL
  14. P. M. Cohn (1975). The Word Problem for Free Fields: A Correction and an Addendum. J. Symbolic Logic 40(1), 69–74. URL
  15. P. M. Cohn (1985). Free Rings and Their Relations. L.M.S. Monographs. Acad. Press. ISBN 9780121791506. URL First edition 1971.
  16. P. M. Cohn (1995). Skew Fields: Theory of General Division Rings. Encyclopedia of Mathematics and its Applications. Cambridge University Press. ISBN 9780521432177. URL
  17. Cohn P.M., Reutenauer C. (1999) On the construction of the free field. International Journal of Algebra and Computation 9(3-4): 307–323MathSciNetCrossRefzbMATHGoogle Scholar
  18. Harm Derksen (2001) Polynomial bounds for rings of invariants. Proceedings of the American Mathematical Society 129(4): 955–964MathSciNetCrossRefzbMATHGoogle Scholar
  19. Harm Derksen & Visu Makam (2015). Polynomial degree bounds for matrix semi-invariants. Preprint arXiv:1512.03393.
  20. Derksen Harm, Weyman Jerzy (2000) Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients. Journal of the American Mathematical Society 13(3): 467–479MathSciNetCrossRefzbMATHGoogle Scholar
  21. M. Domokos, (2000a). Poincaré series of semi-invariants of 2\({\times}\) 2 matrices. Linear Algebra and its Applications 310(1), 183–194.Google Scholar
  22. M. Domokos (2000b). Relative invariants of 3 ×  3 matrix triples. Linear and Multilinear Algebra 47(2), 175–190.Google Scholar
  23. Domokos M. (2002) Finite generating system of matrix invariants. Math. Pannon 13(2): 175–181MathSciNetzbMATHGoogle Scholar
  24. Domokos M., Drensky V. (2012) Defining relation for semi-invariants of three by three matrix triples. Journal of Pure and Applied Algebra 216(10): 2098–2105MathSciNetCrossRefzbMATHGoogle Scholar
  25. Domokos M., Kuzmin S.G., Zubkov A.N. (2002) Rings of matrix invariants in positive characteristic. Journal of Pure and Applied Algebra 176(1): 61–80MathSciNetCrossRefzbMATHGoogle Scholar
  26. Domokos M., Zubkov A.N. (2001) Semi-invariants of quivers as determinants. Transformation groups 6(1): 9–24MathSciNetCrossRefzbMATHGoogle Scholar
  27. Donkin Stephen (1992) Invariants of several matrices. Inventiones mathematicae 110(1): 389–401MathSciNetCrossRefzbMATHGoogle Scholar
  28. Stephen Donkin (1993). Invariant functions on matrices. Mathematical Proceedings of the Cambridge Philosophical Society 113, 23–43. ISSN 1469-8064. URL
  29. Edmonds Jack (1967) Systems of distinct representatives and linear algebra. J. Res. Nat. Bur. Standards Sect. B 71: 241–245MathSciNetCrossRefzbMATHGoogle Scholar
  30. David Eisenbud & Joe Harris (1988). Vector spaces of matrices of low rank. Advances in Mathematics 70(2), 135 – 155. ISSN 0001-8708. URL
  31. Edward Formanek (1986). Generating the ring of matrix invariants. In Ring Theory, Freddy M. J. van Oystaeyen, editor, volume 1197 of Lecture Notes in Mathematics, 73–82. Springer Berlin Heidelberg. ISBN 978-3-540-16496-8. doi: 10.1007/BFb0076314.
  32. M. Fortin & C. Reutenauer (2004). Commutative/Noncommutative Rank of Linear Matrices and Subspaces of Matrices of Low Rank. Séminaire Lotharingien de Combinatoire 52, B52f.Google Scholar
  33. Ankit Garg, Leonid Gurvits, Rafael Oliveira & Avi Wigderson (2015). A deterministic polynomial time algorithm for non-commutative rational identity testing. Preprint arXiv:1511.03730.
  34. Geelen James, Satoru Iwata (2005) Matroid matching via mixed skew-symmetric matrices. Combinatorica 25(2): 187–215MathSciNetCrossRefzbMATHGoogle Scholar
  35. Geelen James F. (1999) Maximum rank matrix completion. Linear Algebra and its Applications 288: 211–217MathSciNetCrossRefzbMATHGoogle Scholar
  36. James F Geelen (2000). An algebraic matching algorithm. Combinatorica 20(1), 61–70Google Scholar
  37. Geelen James F., Iwata Satoru, Murota Kazuo (2003) The linear delta-matroid parity problem. Journal of Combinatorial Theory, Series B 88(2): 377–398MathSciNetCrossRefzbMATHGoogle Scholar
  38. de Graaf Willem A., Ivanyos Gábor, Rónyai Lajos (1996) Computing Cartan subalgebras of Lie algebras. Applicable Algebra in Engineering, Communication and Computing 7(5): 339–349MathSciNetCrossRefzbMATHGoogle Scholar
  39. Gurvits Leonid (2004) Classical complexity and quantum entanglement. J. Comput. Syst. Sci. 69(3): 448–484MathSciNetCrossRefzbMATHGoogle Scholar
  40. Leonid Gurvits & Peter N. Yianilos (1998). The Deflation-Inflation Method for Certain Semidefinite Programming and Maximum Determinant Completion Problems (Extended Abstract). Technical report, NECI.Google Scholar
  41. Nicholas J. A. Harvey, David R. Karger & Kazuo Murota (2005). Deterministic network coding by matrix completion. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, Vancouver, British Columbia, Canada, January 23–25, 2005, 489–498. URL
  42. D. Hilbert (1893). Uber die vollen Invariantensysteme. Math. Ann. (42), 313–370Google Scholar
  43. Pavel Hrubeš & Avi Wigderson (2015). Non-Commutative Arithmetic Circuits with Division. Theory of Computing 11, 357–393. doi: 10.4086/toc.2015.v011a014.
  44. Gábor Ivanyos, Marek Karpinski, Youming Qiao & Miklos Santha (2015a). Generalized Wong sequences and their applications to Edmonds’ problems. J. Comput. Syst. Sci. 81(7), 1373–1386. doi: 10.1016/j.jcss.2015.04.006.
  45. Ivanyos Gábor, Karpinski Marek, Saxena Nitin (2010) Deterministic Polynomial Time Algorithms for Matrix Completion Problems. SIAM J. Comput. 39(8): 3736–3751MathSciNetCrossRefzbMATHGoogle Scholar
  46. Gábor Ivanyos, Youming Qiao & K. V. Subrahmanyam (2015b). Constructive noncommutative rank computation in deterministic polynomial time over fields of arbitrary characteristics. CoRR abs/1512.03531.
  47. Gábor Ivanyos, Youming Qiao & K. V. Subrahmanyam (2015c). On generating the ring of matrix semi-invariants. CoRR abs/1508.01554 .
  48. Kabanets Valentine, Impagliazzo Russell. (2004) Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds. Computational Complexity 13(1–2): 1–46MathSciNetCrossRefzbMATHGoogle Scholar
  49. Erich Kaltofen (1992). On Computing Determinants of Matrices without Divisions. In Proceedings of the 1992 International Symposium on Symbolic and Algebraic Computation, ISSAC ’92, Berkeley, CA, USA, July 27–29, 1992, 342–349. doi: 10.1145/143242.143350.
  50. T.Y. Lam (1991). A First Course in Noncommutative Rings. Graduate Texts in Mathematics. Springer.Google Scholar
  51. Nathan Linial, Alex Samorodnitsky & Avi Wigderson (2000). A Deterministic Strongly Polynomial Algorithm for Matrix Scaling and Approximate Permanents. Combinatorica 20(4), 545–568. doi: 10.1007/s004930070007.
  52. Lovász László (1989) Singular spaces of matrices and their application in combinatorics. Boletim da Sociedade Brasileira de Matemática-Bulletin/Brazilian Mathematical Society 20(1): 87–99MathSciNetCrossRefzbMATHGoogle Scholar
  53. Meena Mahajan & V. Vinay (1997). Determinant: Combinatorics, Algorithms, and Complexity. Chicago Journal of Theoretical Computer Science 1997(5).Google Scholar
  54. Peter Malcolmson (1978). A Prime Matrix Ideal Yields a Skew Field. Journal of the London Mathematical Society s2-18(2), 221–233. URL
  55. Laurent Manivel (2010). A note on certain Kronecker coefficients. Proceedings of the American Mathematical Society 138(1), 1–7Google Scholar
  56. Ketan Mulmuley (1987). A fast parallel algorithm to compute the rank of a matrix over an arbitrary field. Combinatorica 7(1), 101–104. doi: 10.1007/BF02579205.
  57. Ketan Mulmuley (2011). On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna. J. ACM 58(2), 5. doi: 10.1145/1944345.1944346.
  58. Ketan Mulmuley (2012). Geometric Complexity Theory V: Equivalence between Blackbox Derandomization of Polynomial Identity Testing and Derandomization of Noether’s Normalization Lemma. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20–23, 2012, 629–638.  10.1109/FOCS.2012.15.
  59. Ketan Mulmuley & Milind A. Sohoni (2001). Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems. SIAM J. Comput.31(2), 496–526. doi: 10.1137/S009753970038715X.
  60. Ketan Mulmuley & Milind A. Sohoni (2008). Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties. SIAM J. Comput. 38(3), 1175–1206.  10.1137/080718115.
  61. Kazuo Murota (2000). Matrices and matroids for systems analysis. Springer.Google Scholar
  62. Vladimir L Popov (1982). The constructive theory of invariants. Izvestiya: Mathematics 19(2), 359–376Google Scholar
  63. Procesi C. (1976) The invariant theory of n × n matrices. Advances in Mathematics 19(3): 306–381MathSciNetCrossRefzbMATHGoogle Scholar
  64. Ju. P. Razmyslov (1974). Trace identities of full matrix algebras over a field of characteristic zero. Mathematics of the USSR-Izvestiya 8(4), 727. English translation available at
  65. T. G. Room (1938). The Geometry of Determinantal Loci. The Cambridge University Press. URL
  66. Aidan Schofield & Michel Van den Bergh (2001). Semi-invariants of quivers for arbitrary dimension vectors. Indagationes Mathematicae 12(1), 125–138.Google Scholar
  67. Tutte W.T. (1947) The factorization of linear graphs. Journal of the London Mathematical Society 1(2): 107–111MathSciNetCrossRefzbMATHGoogle Scholar
  68. E. Witt (1937). Zyklische Körper und Algebren der Charakteristik p vom Grad pn. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik p. J. Reine Angew. Math 176(01), 126–140.Google Scholar
  69. Kai-Tak Wong (1974). The eigenvalue problem λ Tx + Sx. Journal of Differential Equations 16(2), 270 – 280. ISSN 0022-0396. URL

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Gábor Ivanyos
    • 1
  • Youming Qiao
    • 2
  • K. V. Subrahmanyam
    • 3
  1. 1.Institute for Computer Science and Control, Hungarian Academy of SciencesBudapestHungary
  2. 2.Centre for Quantum Computation and Intelligent SystemsUniversity of Technology SydneySydneyAustralia
  3. 3.Chennai Mathematical InstituteChennaiIndia

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