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Define $\operatorname{ctri}_N:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}^{N\times N}$ by: $${\operatorname{ctri}_N{(a,b)}}_{i,j}=\begin{cases} a, && j>i \\ \frac{a+b}{2}, && j=i \\ b, && j<i \end{cases}$$ for all $a,b\in\mathbb{R}$ and $i,j\in {1,\dots,N}$.

Now, for $X={[x_{ij}]}_{i,j\in{1,\dots,M}}\in\mathbb{R}^{M\times M}$, $Y\in\mathbb{R}^{N\times N}$, define the "symmetrized Kronecker product" (my terminology) by: $$X\bar{\otimes}Y=\begin{bmatrix} Z_{11} && \cdots && Z_{1M} \\ \vdots && \ddots && \vdots \\ Z_{M1} && \cdots && Z_{MM} \end{bmatrix},$$ where for $i,j\in{1,\dots,M}$: $$Z_{i,j}=\begin{cases} \operatorname{ctri}_N{(x_{ij},x_{ji})}\circ Y, && j>i \\ \frac{1}{2}\left[x_{ii}Y+x_{jj}Y'\right], && j=i \\ \operatorname{ctri}_N{(x_{ij},x_{ji})}\circ Y', && j<i \end{cases},$$ where $\circ$ is the usual Hadamard (elementwise) product.

It is easy to see that $X\bar{\otimes}Y$ is always symmetric, and that $Y\bar{\otimes}X$ has the same elements (in a different order) as $X\bar{\otimes}Y$.

My questions are the following: Has this symmetrized Kronecker product been discussed in the literature before? What are its properties?

Note: the "symmetrized Kronecker product" defined here is not equivalent to the usual "symmetric Kronecker product" which is not actually symmetric. See e.g. Schacke (2013) for a definition of the latter.

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