The $n\times n$ matrix $A$ has this specific structure:
- The diagonal entries $a_{ii}$ are strictly increasing and strictly positive real numbers;
- The non-diagonal entries are calculated as $a_{ij}=a_{ji}=a_{ii}$ where $i$ is the smaller index. In other words, $a_{ij}=a_{ji}=\min(a_{ii},a_{jj})$.
Is the matrix $A$ positive-definite?
For the $1\times1$ case, the result is trivial. For the $2\times2$ case, we can use Sylvester's criterion and see that $a_{11}a_{22}-a_{11}^2>0$ is true, so $A$ is positive-definite.
Is there an argument that can be used to extend this result to larger matrices?
Some empirical results
I generated one million $20\times20$ matrices using an exponential random variable for the diagonal. Every random matrix is positive-definite. I expect this result to be true in general but I just don't find a proof.
$a_{ii}=s_1+\cdots+s_i$ implies that
$$\sum_{i,j}a_{i,j}x_ix_j=s_1(x_1+\cdots+x_n)^2+s_2(x_2+\cdots+x_n)^2+\cdots+s_nx_n^2\geq 0.$$