Let $H$ be a Hadamard matrix of order $n$ with first row constant equal to 1. Let $\lambda_1, \ldots, \lambda_n$ be real numbers such that $$\lambda_1 > 0 \geq \lambda_2 \geq \ldots \geq \lambda_n$$ and $$\sum_{i=1}^{n} \lambda_i = 0.$$ We denote by $U$ the matrix $\frac{1}{\sqrt{n}} H$ and $\Lambda$ the diagonal matrix whose diagonal coefficients are the $\lambda_i$. We finally denote by $D = U^T \Lambda U$. Show that $D$ is symmetric, with non-negative coefficients and zero diagonal, and has eigenvalues $\lambda_1, \ldots, \lambda_n$, with $\lambda_1$ having eigenspace of dimension 1.
Let $H$ be a Hadamard matrix of order $n$ with first row constant equal to 1. Let $\lambda_1, \ldots, \lambda_n$ be real numbers such that
$$\lambda_1 > 0 \geq \lambda_2 \geq \ldots \geq \lambda_n$$
and
$$\sum_{i=1}^{n} \lambda_i = 0.$$
We denote by $U$ the matrix $\frac{1}{\sqrt{n}} H$ and $\Lambda$ the diagonal matrix whose diagonal coefficients are the $\lambda_i$. We finally denote by $D = U^T \Lambda U$.
Show that $D$ is symmetric, with non-negative coefficients and zero diagonal, and has eigenvalues $\lambda_1, \ldots, \lambda_n$, with $\lambda_1$ having eigenspace of dimension 1.