Kronecker product based simplification

Question

I am trying to simplify the following block matrix expression based on the Kronecker product: \begin{bmatrix} A \otimes B_{11} & \cdots & A \otimes B_{1m} \\ \vdots\;\;\; & \ddots & \vdots\;\;\; \\[3pt] A \otimes B_{m1} & \cdots & A \otimes B_{mm} \end{bmatrix}

I appreciate any help.

Answer 1

It seems the block matrix you wrote is $B \otimes A$, isn't it?

Answer 2

The specific solution to your query lies within the context of the extended Kronecker product, known as the Tracy Singh product or the block Kronecker product. You can refer to this explicitly in the research paper titled "Block Kronecker Products and the vecb Operator." https://www.sciencedirect.com/science/article/pii/002437959190332Q

Within this paper, you will encounter a thorough definition of the concept you're interested in, presented in Equation (24) as denoted by the operation $A\boxtimes B$, where A remains unpartitioned, and B takes on a specific form.

Furthermore, within the same paper, in Section 2.2, the authors delve into an exploration of the properties of the operator $\boxtimes$. They illustrate that this operator shares many similar attributes with the standard Kronecker product. Notably, the authors establish a connection between the two by demonstrating that the operator $\boxtimes$ can be linked to the standard Kronecker product through the relationship $A\boxtimes B = K_1(B\otimes A)K_2$, where $K_1$ and $K_2$ represent two permutation matrices. Additionally, it's worth noting another noteworthy property of this operation, which states that for vectors ($v$), $v\boxtimes A = A\otimes v$. This property can be a valuable tool in various applications.

If you're keen on delving into the practical applications of this product, you can discover illustrative instances in my recent paper. In this paper, I've utilized this product to derive closed expressions for Intrinsic Local Polynomial estimators on Riemannian Manifolds. You can access these examples by following the link provided below: https://www.researchgate.net/publication/370495044_Multivariate_Intrinsic_Local_Polynomial_Regression_on_Isometric_Riemannian_Manifolds_Applications_to_Positive_Definite_Data

I trust that this provides a satisfactory response to your inquiry.