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