Kronecker factorization

Question

I have found in a paper the following identities (equations $(119)$, $(120)$, appendix D): \begin{equation}\begin{aligned} \sum_{i,j=1}^s p_{ij} d_i d_j'X &= \begin{bmatrix}d_1 \\ \vdots \\ d_s\end{bmatrix}\begin{bmatrix}d_1'X & \cdots &d_s'X\end{bmatrix}\begin{bmatrix}p_{11}I_d & \cdots & p_{1s}I_d \\ \vdots & & \vdots\\ p_{s1}I_d & \cdots & p_{ss}I_d \end{bmatrix}\\ &=\begin{bmatrix}d_1 \\ \vdots \\ d_s\end{bmatrix}\begin{bmatrix}d_1' & \cdots &d_s'\end{bmatrix}(P\otimes X) \end{aligned}\end{equation} where:

1. $d_1$, ..., $d_s$ are column vectors in $\mathbb{R}^{d\times 1}$;
2. $X$ is a symmetric and positively definite square matrix in $\mathbb{R}^{d\times d}$;
3. $I_d$ is the $d\times d$ identity matrix;
4. $P$ is a symmetric and positively definite square matrix in $\mathbb{R}^{s\times s}$ whos structure is \begin{equation}P\triangleq \begin{bmatrix}p_{11}& \cdots & p_{1s}\\ \vdots & & \vdots \\ p_{s1}& \cdots & p_{ss} \end{bmatrix}\end{equation}
5. $\otimes$ denotes the Kronecker product;
6. $'$ denotes the transpose operator (thus $d_i'$ is a row vector in $\mathbb{R}^{1\times d}$).

I'm not sure, but in my opinion these relations are not true: the first member seems to be a summation of $s^2$ different $d\times d$ matrices, while the second member (and also the third member) seems to be a $sd\times sd$ matrix. Thus, the equalities cannot hold.

Answer 1

Indexed vectors are a pain. But here is a simple result that always helps me untangle them.

Let $(A,B)$ be matrices whose columns are $(a_i,b_j)$ respectively, and let $Y$ be an arbitrary matrix with elements $(y_{ij})$ whose dimension are compatible. Then $$\eqalign{ AYB^T = \sum_{i=1}^m\sum_{j=1}^n y_{ij}a_ib_j^T \\ }$$ Apply this to the matrices of the current problem by substituting $A=B=D,\;Y=P,\;$ and $\,m=n=s$. $$\eqalign{ DPD^T = \sum_{i=1}^s\sum_{j=1}^s p_{ij}d_id_j^T \\ }$$ Therefore the entire expression is simply $\,DPD^TX$

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NB: You are correct. The given relationship cannot be correct. What they are proposing is $$ \def\v{{\rm vec}} \Big(\v(D)\otimes\v(D)^T\Big)\big(P\otimes X\big) $$ which is an interesting quantity, but does not match the dimensions of the original sum.