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

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(n2.37) time. Analogously, the maximum order of a square submatrix with nonzero permanent is defined as the permanental rank ρper(A). Computing the permanent or the coefficients of the permanental polynomial per(xI − A) is #P-complete. The permanental nullity ηper(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, ρper(A) + ηper(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. © Priyanshu Pant, Surabhi Chakrabartty, and Ranveer Singh.