The $2^n\times 2^n$ dimensional Hadamard matrices $H_n$ are also called <A HREF="https://en.wikipedia.org/wiki/Hadamard_matrix#Sylvester's_construction">Sylvester matrices</A> or <A HREF="https://en.wikipedia.org/wiki/Walsh_matrix">Walsh matrices.</A> 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 HREF="https://core.ac.uk/download/pdf/81967428.pdf">A note on the eigenvectors of Hadamard matrices of order $2^n$</A> (1982) and in  <A HREF="https://www.sciencedirect.com/science/article/pii/0024379584901307">Some observations on eigenvectors of Hadamard matrices of order $2^n$</A> (1984).