# Row/Column sums 0 matrix is semidefinite positive?

I am reading a paper about colorization via optimization. Let us say that I can construct a symmetric matrix A with the characteristic that the sum of all elements in any row is 0 and all elements on its diagonal are positive. Well, as the sum of all rows of the A matrix is 0, the rows are linearly dependent and then it is not full rank. The A matrix is then singular. For my problem it's pretty obvious that this matrix is semidefinite positive, but I can't proof.

I've tried using the definition xTAx >= 0 without success. Checking at MATLAB, all the eigenvalues are >= 0, so the matrix is semidefinite positive, but I can't proof by this way too. I think that the right way involves the property that the sum of the rows is zero. Some suggestion?

Another question would involve the same semidefinite A matrix. Let C be a diagonal matrix with only a few elements equal to 1. I'm pretty sure that

A + C

is positive definite, and to answer that, I think the first question would be important.

PS: A + C is not diagonally dominant.

So, some suggestion to proof that A is semidefinite positive and A + C is definite positive with those informations?

Thanks.

$$\left( \begin{array}{rrr} 1 & 2 & -3 \\ 2 & 1 & -3 \\ -3 & -3 & 6 \end{array} \right) ,$$
Eigenvalues are $9,0,-1.$ Eigenvector $(1,-1,0)$ has eigenvalue $-1.$