Let $A$ be a real symmetric $3\times 3$ matrix with entries belonging to the set of non-negative integers. Also assume that each diagonal entries are greater than or equal to all the entries in the row and column in which the diagonal entry belongs.
I want to know whether all the eigenvalues of the matrix is non-negative, i.e., it is a positive semidefinite matrix or not.
Note that this is true if we consider the $2\times 2$ matrices of the above type. I just want to know whether this is true for $3\times 3$ case.
Best Answer
It's not true. The following matrix, for example, has a negative determinant (and therefore a negative eigenvalue). $$\begin{bmatrix}1&1&0\\ 1&2&2\\ 0&2&2\end{bmatrix}$$
Its determinant is $-2$.