A (±1)-matrix is a matrix whose entries are 1 and −1. An $n \times n$ (±1)-matrix is called an Hadamard matrix if the rows are orthogonal.

Equivalently, An $n \times n$ (±1)-matrix $H$ is Hadamard ⇔ $H H^t = nI_n$, where $I_n$ denotes the $n \times n$ identity matrix.

In this paper : http://www.sciencedirect.com/science/article/pii/0024379582902105

A note on the eigenvectors of Hadamard matrices of order $2^n$ R. Yarlagadda J. Hershey

we have the result that, If the order of $H$, is $2^n$, then its $2^n$ eigenvalues as follows:

$2^{n - 1}$ eigenvalues are $2^{\frac{n}{2}}$

$2^{n - 1}$ eigenvalues are $- 2^{\frac{n}{2}}$

This result says that $H_n$ has $\frac{n}{2}$ eigen values equal to $n^{\frac{1}{2}}$ and $\frac{n}{2}$ eigenvalues equalt to $- n^{\frac{1}
{2}}$ if **$n = 2^n$**.

My questions are,

1) Is this result true for any Hadamard matrix of order $n$ (this $n$ is any multiple of 4 for which we know there is a Hadamard matrix of that order) ?

2) What is best known result regarding their eigen values?

Thanks for your valuable timing.