The algorithm for the symbolic computation of the Aluthge transform of a polynomial matrix is derived in this paper. For this purpose, the well-known PSVD by PQRD algorithm is considered to avoid square roots of polynomials in the Aluthge transform matrix. The algorithm for the symbolic computation of the polar decomposition for polynomial matrices is developed as well. Furthermore, the Aluthge transform for rank-one matrices defined by cross-products of vectors is symbolically calculated and presented in a closed formula.