# Expanding a product of matrices with tensor product and transpose

I'm trying to expand the following product of $\pm1$ matrices $H_1, H_2, K_1, K_2$: $(\frac12 (H_1+H_2)\otimes K_1^T+\frac12 (H_1-H_2)\otimes K_2^T)(\frac12 (H_1+H_2)\otimes K_1^T+\frac12 (H_1-H_2)\otimes K_2^T)^T$ These matrices come from the paper 'Product of four matrices' which I'm trying to understand.

My idea is: $(\frac12 (H_1+H_2)\otimes K_1^T+\frac12 (H_1-H_2)\otimes K_2^T)^T = (\frac12 (H_1+H_2)^T\otimes K_1+\frac12 (H_1-H_2)^T\otimes K_2) = (\frac12 (H_1^T+H_2^T)\otimes K_1+\frac12 (H_1^T-H_2^T)\otimes K_2)$.

Multiplying the first terms gives : $(\frac12 (H_1+H_2)\otimes K_1^T)(\frac12 (H_1^T+H_2^T)\otimes K_1)= \frac12 (H_1H_1^T+ H_1H_2^T+ H_2H_1^T+ H_2H_2^T)\otimes K_1^T K_1$.

But I fail to see how this can simplified any more. What possibilities are there to simplify the expression?

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