A constructive algorithm to compute elimination \bar {L} and duplication \bar {D} matrices for the operation of P \otimes P vectorization when P = {{P}^{{\text{T}}}} is proposed. The matrix \bar {L} , obtained according to such algorithm, allows one to form a vector that contains only unique elements of the mentioned Kronecker product. In its turn, the matrix \bar {D} is for the inverse transformation. A software implementation of the procedure to compute the matrices \bar {L} and \bar {D} is developed. On the basis of the mentioned results, a new operation {\text{vecu}}\left(. \right) is defined for P \otimes P in case P = {{P}^{{\text{T}}}} and its properties are studied. The difference and advantages of the developed operation in comparison with the known ones {\text{vec}}\left(. \right) and {\text{vech}}\left(. \right) ({\text{vecd}}\left(. \right)) in case of vectorization of P \otimes P when P = {{P}^{{\text{T}}}} are demonstrated. Using parameterization of the algebraic Riccati equation as an example, the efficiency of the operation {\text{vecu}}\left(. \right) to reduce overparameterization of the unknown parameter identification problem is shown.