Let $n \times n$ matrix $M(n):=(m_{ij})$ be defined by $$m_{ij} = \min(i,j)$$
This matrix enumerates certain combinatorial objects. Do you know any interesting properties of matrix $M$?
Let $n \times n$ matrix $M(n):=(m_{ij})$ be defined by $$m_{ij} = \min(i,j)$$
This matrix enumerates certain combinatorial objects. Do you know any interesting properties of matrix $M$?
Let $A=(a_{ij})$ be the upper triangular matrix such that $a_{ij}= \begin{cases} 1, \textrm{if $i \leq j$}\\ 0, \textrm{else} \end{cases}$, then the matrix $M$ is equal to the product $A^{T}A$. Many interesting properties of $M$ follow from this.
For example, one concludes that $M$ has determinant $1$, and is positive definite. What else do you want to know?