The Complexity of the Characteristic and the Minimal Polynomial

Thanh Minh Hoang and Thomas Thierauf

Abstract:
We investigate the complexity of
  1. computing the characteristic polynomial, the minimal polynomial, and all the invariant factors of an integer matrix, and of
  2. verifying them, when the coefficients are given as input.
It is known that each coefficient of the characteristic polynomial of a matrix A is computable in GapL, and the constant term, the determinant of A, is complete for GapL. We show that the verification of the characteristic polynomial is complete for complexity class C=L (exact counting logspace).

We show that each coefficient of the minimal polynomial of a matrix A can be computed in AC0(GapL) the AC0-closure of GapL, and there is a coefficient which is hard for GapL. Furthermore, the verification of the minimal polynomial is in AC0(C=L) and is hard for C=L. The hardness result extends to (computing and verifying) the system of all invariant factors of a matrix.