Given a symmetric matrix [tex]A=\left(\begin{array}{ccccc} \sum a_{1s} & & & & \\ & \ddots & & a_{ij} \\ & & \ddots & & \\ &a_{ij} & & \ddots & \\ & & & & \sum w_{as} \end{array}\right) \in\mathbb{R}^{n\times n}, [/tex] with strictly positive entries a_{ij}, and with the diagonal entries being sum of off-diagonal entries residing in the corresponding row/column, how to proceed with the proof for A being positive definite, [tex] x^TAx>0 [/tex] for some non-zero vector x.
The http://en.wikipedia.org/wiki/Gershgorin_circle_theorem shows there are no negative eigenvalues, but it doesn't exclude the possibiltiy of zero eigenvalues (i.e. a singular matrix). In fact the matrix $$\begin{pmatrix}1 & 1 \\ 1 & 1 \end{pmatrix}$$ is singular, and therefore not positive definite.
Thanks for providing the example. I guess the author of the book stating the above positive-definiteness on the given matrix type somehow misinterpreted it.
This is true because the matrix is diagonally dominant. There is a theorem that says a Hermitian diagonally dominant matrix with real nonnegative diagonal entries is positive semidefinite. A proof of this is found here http://planetmath.org/?op=getobj&from=objects&id=7483