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We investigate a method of the full-rank LDL$^*$ factorization, where matrices $L$ and $D$ are abbreviated by zero rows and columns. Therefore, we provide explicit formulaes for the coefficients appearing in rational matrices $L$ and $D$. An algorithm for the computation of $A_{T,S}^{(2)}$ inverses of a given matrix $A$, based on the full--rank $LDL^*$ decomposition of an appropriate matrix $M$, is derived. This method is extended to the set of one--variable polynomial matrices, aiming to efficiently compute $A_{T,S}^{(2)}$ inverses of polynomial Hermitian matrices. We explain the implementation details in programming language {\ssr MATHEMATICA}, illustrate our algorithms via examples and compare them to other well-known methods.

LDL$^*$ decomposition, {\ssr MATHEMATICA\/}, polynomial matrix, Hermitian matrix, generalized inverse.