$\lambda_{\min}$ and $\lambda_{\max}$ of rank-1 sum of matrices

Question

It is explained from previous posts^1,2 that for a rank-$1$ matrix $x_ix_i^T$ we have $\lambda_{\max} (x_ix_i^T)=1$ and $\lambda_{\min} (x_ix_i^T)=0$ with single and $N-1$ algebraic multiplicity, respectively.

Could you please provide some help to compute the $\lambda_{\min}(A)$ and $\lambda_{\max}(A)$ of $$A=\sum_{j=1}^{n}x_ix_j^T$$ with $x_j\in\mathbb{R}^{N}$, $x_j^Tx_j=1$, $x_i^Tx_j=0, i\neq j$ and $n < N$.

Answer 1

Still $0$ and $1$. Your $A$ is a projection: $A^2=A$, so its eigenvalues need to satisfy $\lambda^2=\lambda $; thus the only possible eigenvalues are $0$ and $1$. And both eigenvalues are realized, since $Ax_1=x_1$, and $Ax_N=0$.