non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix A=(aij)i,j=1,,dRd×d with the following property: aii=jiaij, 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 BRdk×dk, CRk×k, k{1,,d1}, RGL(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:

  1. Is A positive semi-definite in the sense that xTAx0 for all xRd{0}?
  2. By the property of the summation of each row to zero, it is immediately clear that (1,,1)Tker(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

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