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A Permanental Analog of the Rank-Nullity Theorem for Symmetric Matrices

Authors Priyanshu Pant , Surabhi Chakrabartty , Ranveer Singh

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LIPIcs.STACS.2026.70.pdf
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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Combinatorics
  • Theory of computation → Complexity theory and logic
Keywords
  • permanent
  • matrix rank
  • #P-completeness
  • graph algorithms
  • permanental polynomial
  • spectral graph theory

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Abstract

The rank of an n × n matrix A is equal to the maximum order of a square submatrix with a nonzero determinant; it can be computed in O(n^{2.37}) time. Analogously, the maximum order of a square submatrix with nonzero permanent is defined as the permanental rank ρ_{per}(A). Computing the permanent or the coefficients of the permanental polynomial per(xI-A) is #P-complete. The permanental nullity η_{per}(A) is defined as the multiplicity of zero as a root of the permanental polynomial. We establish a permanental analog of the rank–nullity theorem, ρ_{per}(A) + η_{per}(A) = n for symmetric nonnegative matrices, positive semidefinite matrices, and adjacency matrices of balanced signed graphs. Using this theorem, we can compute the permanental nullity for these classes in polynomial time. For {0,± 1}-matrices, we also provide a complete characterization of when the permanental rank-nullity identity holds.

Cite As Get BibTex

Priyanshu Pant, Surabhi Chakrabartty, and Ranveer Singh. A Permanental Analog of the Rank-Nullity Theorem for Symmetric Matrices. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 70:1-70:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.70

Author Details

Priyanshu Pant
  • Department of Computer Science and Engineering, Indian Institute of Technology Indore, India
Surabhi Chakrabartty
  • Department of Mathematical Sciences, Indian Institute of Science Education and Research Berhampur, India
Ranveer Singh
  • Department of Computer Science and Engineering, Indian Institute of Technology Indore, India

Funding

  • Pant, Priyanshu: Funded by CSIR-UGC NET JRF Fellowship.
  • Singh, Ranveer: Supported by DST INSPIRE Grant.

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