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Constructive Algorithm to Vectorize P ⊗ P Product for Symmetric Matrix P
- Authors: Glushchenko A.I.1, Lastochkin K.A.1
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Affiliations:
- V.A. Trapeznikov Institute of Control Sciences of RAS
- Issue: Vol 63, No 9 (2023)
- Pages: 1415-1427
- Section: General numerical methods
- URL: https://ogarev-online.ru/0044-4669/article/view/136176
- DOI: https://doi.org/10.31857/S0044466923090090
- EDN: https://elibrary.ru/ROMVVI
- ID: 136176
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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.
About the authors
A. I. Glushchenko
V.A. Trapeznikov Institute of Control Sciences of RAS
Email: aiglush@ipu.ru
117997, Moscow, Russia
K. A. Lastochkin
V.A. Trapeznikov Institute of Control Sciences of RAS
Author for correspondence.
Email: lastconst@ipu.ru
117997, Moscow, Russia
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