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Certificates to a linear algebra computation are additional data structuresfor each output, which can be used by a-possibly randomized- verificationalgorithm that proves the correctness of each output. Wiede-mann's algorithmprojects the Krylov sequence obtained by repeatedly multiplying a vector by amatrix to obtain a linearly recurrent sequence. The minimal polynomial of thissequence divides the minimal polynomial of the matrix. For instance, if the$n\times n$ input matrix is sparse with n 1+o(1) non-zero entries, thecomputation of the sequence is quadratic in the dimension of the matrix whilethe computation of the minimal polynomial is n 1+o(1), once that projectedKrylov sequence is obtained. In this paper we give algorithms that computecertificates for the Krylov sequence of sparse or structured $n\times n$matrices over an abstract field, whose Monte Carlo verification complexity canbe made essentially linear. As an application this gives certificates for thedeterminant, the minimal and characteristic polynomials of sparse or structuredmatrices at the same cost.

Interactive certificate for the verification of Wiedemann's Krylov  sequence: application to the certification of the determinant, the minimal  and the characteristic polynomials of sparse matrices