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The $2^n\times 2^n$ dimensional Hadamard matrices $H_n$ are also called Sylvester matrices or Walsh matrices. 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 $2^n$ (1982) and in Some observations on eigenvectors of Hadamard matrices of order $2^n$ (1984).

The $2^n\times 2^n$ dimensional Hadamard matrices $H_{2^n}$ are also called Sylvester matrices or Walsh matrices. 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 $2^n$ (1982) and in Some observations on eigenvectors of Hadamard matrices of order $2^n$ (1984). See also Chapter 5 of Hadamard Matrix Analysis and Synthesis (2012).

The matrices $H_n$ are also called Walsh matrices. There are only two distinct eigenvalues $\pm 2^{n/2}$, so the eigenvectors are not in general orthogonal. An orthogonal basis is constructed recursively in A note on the eigenvectors of Hadamard matrices of order $2^n$ (1982) and in Some observations on eigenvectors of Hadamard matrices of order $2^n$ (1984).

The $2^n\times 2^n$ dimensional Hadamard matrices $H_n$ are also called Sylvester matrices or Walsh matrices. 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 $2^n$ (1982) and in Some observations on eigenvectors of Hadamard matrices of order $2^n$ (1984).

These are also called Walsh matrices. For the eigenvectors, see A note on the eigenvectors of Hadamard matrices of order $2^n$ There are only two distinct eigenvalues $\pm 2^{n/2}$ of $H_n$, so the eigenvectors are not in general orthogonal. An orthogonal basis is constructed in the cited paper.

The matrices $H_n$ are also called Walsh matrices. There are only two distinct eigenvalues $\pm 2^{n/2}$, so the eigenvectors are not in general orthogonal. An orthogonal basis is constructed recursively in A note on the eigenvectors of Hadamard matrices of order $2^n$ (1982) and in Some observations on eigenvectors of Hadamard matrices of order $2^n$ (1984).