Let $M \in \mathbb{R}^{m \times n}$ be a matrix of rank $r$. Let $U\Sigma V^T$ be a compact SVD of $M$ ($U \in \mathbb{R}^{m \times r}$, $\Sigma=\operatorname{diag}(\sigma_1,\dots,\sigma_r)$ with $\sigma_1 \geq \dots \sigma_r > 0$, $V \in \mathbb{R}^{n \times r}$). We say that $M$ is homogeneous if :
- $m=\Theta(n)$
- $r=\Theta(1)$
- $\left\| M\right\|_{\infty}=\Theta(1)$
- $\frac{\sigma_1}{\sigma_r}=\Theta(1)$
- $\mu=\max(\sqrt{m/r}\left\|U\right\|_{2 \to \infty},\sqrt{n/r}\left\|V\right\|_{2 \to \infty})=\Theta(1)$
Here the notation $a(n,m,r)=\Theta(b(n,m,r))$ means that there are positive constants $c,C$ such that $ca(n,m,r)\geq b(n,m,r) \geq Ca(n,m,r)$ for every $n,m,r$. The two-to-infinity norm of a matrix is the largest 2-norm of the rows of the matrix.
I am looking for simple examples of homogeneous matrices. The matrix with all $1$'s is an example when $r=1$, but I'd like to have an example for an arbitrary $r$.