Let $L$ be a Laplacian matrix of a strong connected and balanced directed graph. Define $$ L^{s}=\frac{1}{2}\left( L+L^{T}\right) .$$ Let $D$ be a diagonal matrix with $$ D=\begin{bmatrix} d_{1} & & & \\ & d_{2} & & \\ & & \ddots & \\ & & & d_{n}% \end{bmatrix}, $$ with $d_{i}\geq 0.$ There is at least one $d_{i}>0$. Clearly, this matrix is positive semi-definite. Is the matrix $ L^{s}+D $ positive definite or not?
How to prove that a matrix is positive definite?
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linear-algebra
matrices
graph-theory
algebraic-graph-theory
topological-graph-theory
1 Answers
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Take any nonzero $\mathbf{x}\in\mathbf{R}^n$ and let $\mathbf{x}=(x_1,x_2,\dotsc ,x_n)$.
Then
$\quad \mathbf{x} L_s\mathbf{x}^t=\sum_{ij} (x_i-x_j)^2 > 0$,
where the sum is taken over the edges of the graph (edges without orientation), which shows that $L_s$ is positive definite (the graph is strongly connected). On the other hand $D$ is clearly positive semi-definite, and hence the addition of both matrices is positive definite.
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1Thanks for A.Schulz's answer. I got some hint from you. The matrix $L^{s}+D$ is positive definite, not just positive semi-definite. Follow A.Schulz's proof, we have $ x\left( L^{s}+D\right) x=\sum_{ij}\left( x_{i}-x_{j}\right) ^{2}+\sum_{i}d_{i}x_{i}^{2} $. Suppose $d_{i}$ is positive. If $x\left( L^{s}+D\right) x=0$, then $x_{i}=0$, then the graph is strongly connected, then all $x_{j}=0$. Thus we get that $L^{s}+D$ is positive definite. – 2012-07-22
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0@XiangyuMeng: Thanks, I have missed the fact that the one of the $d_i$s is positive. I edited the answer. – 2012-07-22
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0I don't know exactly what you missed (that at least one of the $d_i$ is positive, or that all the others can be zero), but in any case $D$ is only positive semi-definite; it is definitely :-) not "clearly positive definite". – 2012-07-22
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0@MarcvanLeeuwen Thanks for the pointer. – 2012-07-22
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0We cannot allow that the graph contains no edges at all since the graph is strongly connected. – 2012-07-22
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0@XiangyuMeng: Darn, I totally missed that. I was getting your first answer wrong, and was only looking at the restrictions of $D$. Thanks! – 2012-07-23