New facts related to dilation factorizations of Kronecker products of matrices

Abstract

The Kronecker product of two matrices is known as a special algebraic operation of two arbitrary matrices in the computational aspect of matrix theory. This kind of matrix operation has some interesting and striking operation properties, one of which is given by $ (A \otimes B)(C \otimes D) = (AC) \otimes (BD) $ and is often called the mixed-product equality. In view of this equality, the Kronecker product $ A_1 \otimes A_2 $ of any two matrices can be rewritten as the dilation factorization $ A_1 \otimes A_2 = (A_1 \otimes I_{m_2})(I_{n_1} \otimes A_2) $, and the Kronecker product $ A_1 \otimes A_2 \otimes A_3 $ can be rewritten as the dilation factorization $ A_1 \otimes A_2 \otimes A_3 = (A_1\otimes I_{m_2} \otimes I_{m_3})(I_{n_1} \otimes A_2 \otimes I_{m_3})(I_{n_1} \otimes I_{n_2} \otimes A_3) $. In this article, we proposed a series of concrete problems regarding the dilation factorizations of the Kronecker products of two or three matrices, and established a collection of novel and pleasing equalities, inequalities, and formulas for calculating the ranks, dimensions, orthogonal projectors, and ranges related to the dilation factorizations. We also present a diverse range of interesting results on the relationships among the Kronecker products $ I_{m_1} \otimes A_2 \otimes A_3 $, $ A_1 \otimes I_{m_2} \otimes A_3 $ and $ A_1 \otimes A_2 \otimes I_{m_3} $.

Keywords

• dilation factorization
• dimension
• kronecker product
• orthogonal projector
• range
• rank