Computing the Frobenius Normal Form of a Sparse Matrix

We probabilistically determine the Frobenius form and thus the characteristic polynomial of a matrix $$A \in {^{n \times n}}$$ by O(μnlog(n)) multiplications of A by vectors and 0 (μn2 log2 (n)loglog(n)) arithmetic operations in the field F . The parameter μ.L is the number of distinct invariant factors of A, it is less than $$3{{\sqrt n } \mathord{\left/ {\vphantom {{\sqrt n } 2}} \right. \kern-\nulldelimiterspace} 2}$$in the worst case. The method requires O(n) storage space in addition to that needed for the matrix A.