The complexity of verifying the characteristic polynomial and testing similarity

We investigate the computational complexity of some important problems in linear algebra. 1. The problem of verifying the characteristic polynomial of a matrix is known to be in the complexity class C₌L (Exact Counting in Logspace). We show that it is complete for C₌L under logspace many-one reductions. 2. The problem of deciding whether two matrices are similar is known to be in the complexity class AC⁰(C=L). We show that it is complete for this class under logspace many-one reductions. We also consider the problems of deciding equivalence and congruence of matrices