Can we call a symmetric matrix with
- diagonal entries all zero and
- off-diagonal entries greater than or equal to zero,
a hollow matrix?
Another way around: Can a hollow matrix have off-diagonal entries equal to zero (at least one entry)? Or, is that matrix called in a different way?
For example, are both these two matrices, hollow matrices?
$$\begin{vmatrix} 0 & 5 & 0 & 2 \\ 5 & 0 & 6 & 1 \\ 0 & 6 & 0 & 3 \\ 2 & 1 & 3 & 0 \end{vmatrix}, \qquad \begin{vmatrix} 0 & 5 & 1 & 2 \\ 5 & 0 & 6 & 1 \\ 1 & 6 & 0 & 3 \\ 2 & 1 & 3 & 0 \end{vmatrix}$$
Yes, the two matrices in your question are hollow. However, the definition of a hollow matrix only requires the matrix to be a square one with a zero diagonal. This definition applies to square matrices over any field. It does not require symmetry or entrywise non-negativity. In fact, the values of the off-diagonal entries or the structure of the off-diagonal part are completely irrelevant. As long as the matrix is square and all of its main diagonal elements are zero, it can be called a hollow matrix.