Are there symmetric positive definite matrices whose diagonal entries are not the largest entries in their row/column in absolute value?
That property is weaker than diagonal dominance (say, rowwise) where the diagonal entries are larger than the sums of the absolute values of the other row entries. It is clear that there are symmetric positive definite matrices that are not diagonally dominant. However, I do not know whether the above weaker property does have exceptions among s.p.d. matrices.
Best Answer
The matrix $\pmatrix{5&2\\2&1}$ is symmetric positive definite but the diagonal entry $1$ is not the largest in its row or column.