We are considering a matrix A=(aij)i,j=1,…,d∈Rd×d with the following property: aii=−∑j≠iaij, i.e., the matrix is not only weak diagonal-dominant, but its rows also sum up to 0. Note that the matrix is not necessarily symmetric (otherwise positive semi-definiteness would follow immediately from the weak diagonal-dominance). Furthermore, the matrix has the property that there are no matrices B∈Rd−k×d−k, C∈Rk×k, k∈{1,…,d−1}, R∈GL(d,R) such that

A=R⋅(B00C)⋅RT,

i.e., we can not rearrange the matrix A by switching two rows and the corresponding columns to obtain a decoupled version of A. Two questions arise:

- Is A positive semi-definite in the sense that xTAx≥0 for all x∈Rd∖{0}?
- By the property of the summation of each row to zero, it is immediately clear that (1,…,1)T∈ker(A). Is dimker(A)=1 or is there another, linearly independent element?
These questions arose when considering the following paper: “Constructing Laplace Operator from Point Clouds in Rd“, by Belkin, Sun, and Wang, 2009, where the properties (1) and (2) are claimed for a matrix formulation of a discretization of the Laplace operator, but are not proved there and do not seem trivial to me.

**Answer**

**Attribution***Source : Link , Question Author : Skrodde , Answer Author : Community*