Permanental Analog of the Rank-Nullity Theorem for Symmetric Matrices
Priyanshu Pant, Surabhi Chakrabartty, Ranveer Singh
TL;DR
This work introduces a permanental analog of the rank-nullity theorem by studying the permanental rank $\\rho_{per}(A)$ and permanental nullity $\\eta_{per}(A)$ of symmetric matrices via the permanental polynomial $\\pi(A,x)$. It proves a universal inequality $\\rho_{per}(A)+\\eta_{per}(A)\\ge n$ and identifies a condition $E_k \\\neq O_k$ (with $k=\\rho_{per}(A)$) that characterizes equality, connecting cycle covers and Sachs subgraphs to this equality. The authors show the equality holds for nonnegative symmetric, positive semidefinite, and adjacency matrices of balanced signed graphs, enabling polynomial-time computation of the permanental nullity in these cases, and provide a complete criterion for $\\{0,\\pm1\\}$-symmetric matrices. The results illuminate tractable aspects of permanental structure in specific matrix classes and motivate a broader theory linking permanents with graph structure and algorithmic properties.
Abstract
The rank of an n x n matrix A is equal to the size of its largest square submatrix with a nonzero determinant, and it can be computed in O(n^2.37) time. Analogously, the size of the largest square submatrix with nonzero permanent is defined as the permanental rank. Computing the permanent or the coefficients of the permanental polynomial is #P-complete. The permanental nullity is defined as the multiplicity of zero as a root of the permanental polynomial. We establish a permanental analog of the rank-nullity theorem, showing that the sum of the permanental rank and the permanental nullity equals 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 symmetric nonnegative matrices and adjacency matrices of balanced signed graphs in polynomial time. For symmetric matrices with entries in {0, plus or minus 1}, we also provide a complete characterization of when the permanental rank-nullity identity holds.