What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix defined recursively by $H\_1=(1)$ and $$ H\_N=\begin{pmatrix}H\_{N/2} & H\_{N/2} \\ H\_{N/2} & -H\_{N/2}\end{pmatrix}, $$ where $N=2^n$?


Edit: The answer below provides a "literal" answer to the problem. However, is there a deeper meaning to the eigenvectors? For the Fourier transform operator, for example, Hermite polynomials provide an excellent and rich theory of the eigenvectors. Since the Hadamard transform is indeed a Fourier transform (over the Boolean cube as the underlying group), one could expect the eigenvectors to have a clean interpretation.



What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix?
The $2^n\times 2^n$ dimensional Hadamard matrices $H\_{2^n}$ are also called [Sylvester matrices](https://en.wikipedia.org/wiki/Hadamard_matrix#Sylvester%27s_construction) or [Walsh matrices.](https://en.wikipedia.org/wiki/Walsh_matrix) There are only two distinct eigenvalues $\pm 2^{n/2}$, so the eigenvectors are not in general orthogonal. An orthogonal basis of eigenvectors is constructed recursively in [A note on the eigenvectors of Hadamard matrices of order $2^n$](https://core.ac.uk/download/pdf/81967428.pdf) (1982) and in [Some observations on eigenvectors of Hadamard matrices of order $2^n$](https://www.sciencedirect.com/science/article/pii/0024379584901307) (1984). See also Chapter 5 of [Hadamard Matrix Analysis and Synthesis](https://books.google.nl/books?id=qwrTBwAAQBAJ) (2012).


