The full-rank $LDL^*$ decomposition of a polynomial Hermitian matrix is examined. Explicit formulae are given evaluating the coefficients of matrices $l_{ij}$ and $l_{jj}$. Also, a new method is developed, based on the $LDL^*$ factorization of the matrix product $A^*A$, for symbolic computation of the Moore--Penrose inverse matrix. The paper follows the results of [I. P. Stanimirovi\'c, M. B. Tasi\'c, Computation of generalized inverses by using the $LDL^*$ decomposition, Appl. Math. Lett., 25 (2012), 526--531], where the matrix products $A^*A$, $AA^*$ and the corresponding $LDL^*$ factorizations are considered in order to compute the generalized inverse of $A$.