MHB Eigenvalues of Laplacian are non-negative

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The discussion focuses on proving that for a vector x in R^n, the equation x.Lx=0.5 sum(Aij(xi-xj)^2 holds true, where A is the adjacency matrix and L is the Laplacian. The user seeks assistance in understanding this proof and its implications for demonstrating that the eigenvalues of L are non-negative for indices 1<j<n. The importance of posting questions in the correct subforum to avoid duplication of effort is also emphasized. The community aims to provide valuable help efficiently. Overall, the thread highlights a mathematical inquiry related to graph theory and eigenvalue properties.
tarnat
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Hi, I need to learn the following proof and I'm having trouble getting my head round it. Any help would be appreciated.

Show that if vector x in R^n with components x=(x1,x2,...,xn), then
x.Lx=0.5 sum(Aij(xi-xj)^2)
where A is the graphs adjacency matrix, L is laplacian.
Then use this result to prove that the eigen values of L are non-zero for all 1<j<n.

Thanks.
 
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I have deleted the duplicate of this thread posted in the Discrete Mathematics subforum.

We ask that a question be posted only once and in the appropriate subforum. This eliminates the possibility of duplication of effort on the part of our helpers, whose time is valuable. :D
 
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Thread 'How to define a vector field?'

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