Constructive Algorithm to Vectorize P ⊗ P Product for Symmetric Matrix P

Abstract

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.