Suppose we have a $d\times d$ matrix with $k$'th row consisting of $k$ constants $\underbrace{c, c,\ldots,c}_{k}$ and the rest 0. Rows are normalized to have norm 1. Can we say anything about singular values of this matrix?
IE, for $d=5$ $$\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0 & 0 \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & 0 & 0 \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{5}} \\ \end{array} \right)$$
Empirically, it appears that norm $\approx \sqrt{d}$ and the rest of the singular values decay approximately as $O(1/k)$ for large $d$. Meanwhile $k$th eigenvalues is exactly $\frac{1}{\sqrt{k}}$. Is there an easy way to explain this?
