Constructive algorithm to vectorize $P\otimes P$ product for symmetric matrix $P$

A constructive algorithm to compute elimination $\bar L$ and duplication $\bar D$ matrices for the operation of $P\otimes P$ vectorization when $P=P^{\mathrm{T}}$ 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 $\mathrm{vecu}(.)$ is defined for $P\otimes P$ in case $P=P^{\mathrm{T}}$ and its properties are studied. The difference and advantages of the developed operation in comparison with the known ones $\mathrm{vec}(.)$ and $\mathrm{vech}(.)$ $\mathrm{vecd}(.)$ in case of vectorization of $P\otimes P$ when $P=P^{\mathrm{T}}$ are demonstrated. Using parameterization of the algebraic Riccati equation as an example, the efficiency of the operation $\mathrm{vecu}(.)$ to reduce overparameterization of the unknown parameter identification problem is shown.