# Matrix Polynomial Factorization via Higman Linearization

In continuation to our recent work on noncommutative polynomial factorization, we consider the factorization problem for matrices of polynomials and show the following results. (1) Given as input a full rank d× d matrix M whose entries M_ij are polynomials in the free noncommutative ring 𝔽_q⟨ x_1,x_2,…,x_n ⟩, where each M_ij is given by a noncommutative arithmetic formula of size at most s, we give a randomized algorithm that runs in time polynomial in d,s, n and log_2q that computes a factorization of M as a matrix product M=M_1M_2⋯ M_r, where each d× d matrix factor M_i is irreducible (in a well-defined sense) and the entries of each M_i are polynomials in 𝔽_q ⟨ x_1,x_2,…,x_n ⟩ that are output as algebraic branching programs. We also obtain a deterministic algorithm for the problem that runs in poly(d,n,s,q). (2)A special case is the efficient factorization of matrices whose entries are univariate polynomials in 𝔽[x]. When 𝔽 is a finite field the above result applies. When 𝔽 is the field of rationals we obtain a deterministic polynomial-time algorithm for the problem.

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