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

Abstract

A constructive algorithm to compute elimination L and duplication D matrices for the operation of P ⊗ P vectorization when P = PT is proposed. The matrix 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 D is for the inverse transformation. A software implementation of the procedure to compute the matrices L and D is developed. On the basis of the mentioned results, a new operation vecu(.) is defined for P ⊗ P in case P = PT and its properties are studied. The difference and advantages of the developed operation in comparison with the known ones vec(.) and vech(.) vecd(.)) in case of vectorization of P ⊗ P when P = PT are demonstrated. Using parameterization of the algebraic Riccati equation as an example, the efficiency of the operation vecu (.) to reduce overparameterization of the unknown parameter identification problem is shown.