# Can the Kronecker product for the Hadamard matrix with itself be written as a matrix multiplication?

The Hadamard is a two by two matrix:

$$$$\frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \\ \end{bmatrix}$$$$

The Kronecker product of the Hadamard with itself is $$H \otimes H$$ or $$H^{\otimes 2}$$

Let X be a $$2 \times 4$$ complex valued matrix, and let $$\overline{X}$$ be its conjugate transpose.

Is it possible to write $$H^{\otimes 2}$$ as $$\overline{X} H X$$ ?

More generally, is there a way to find a $$4 \times 8$$ matrix $$Y$$ such that $$H^{\otimes 3} = \overline{Y} \space\overline{X} H X \space Y$$

and so on for higher order Kronecker products of the Hadamard?

$$H\otimes H$$ has rank $$4$$, but $$\operatorname{rank}(PHQ)\le\operatorname{rank}(H)=2$$.