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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.