Eigenvalues of Laplacian are non-negative

In summary, the conversation discusses a proof involving a vector x in R^n and its components, as well as the adjacency matrix A and Laplacian L. The proof shows that x.Lx is equal to 0.5 times the sum of Aij multiplied by (xi-xj)^2. This result is then used to prove that the eigen values of L are non-zero for all values of j between 1 and n.
  • #1
tarnat
1
0
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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  • #2
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
 

FAQ: Eigenvalues of Laplacian are non-negative

What is the Laplacian operator?

The Laplacian operator is a mathematical operator that appears in several areas of physics and mathematics, including differential equations and vector calculus. It is commonly denoted by the symbol ∇² and is defined as the sum of the second partial derivatives of a function with respect to each of its variables.

What are eigenvalues?

Eigenvalues are a set of numbers associated with a linear transformation or matrix. They represent the scalar values by which certain vectors are scaled when the transformation is applied to them. In other words, they are the values that remain unchanged when a linear transformation is applied to a vector.

Why are the eigenvalues of the Laplacian operator important?

The eigenvalues of the Laplacian operator are important because they provide information about the behavior of solutions to differential equations. In particular, the eigenvalues of the Laplacian are closely related to the stability and convergence of solutions to certain differential equations.

What does it mean for eigenvalues of the Laplacian to be non-negative?

If the eigenvalues of the Laplacian are non-negative, it means that all of the eigenvalues have values greater than or equal to zero. In other words, all of the eigenvalues are positive or zero. This property is important in many applications, as it ensures that certain solutions to differential equations are stable and well-behaved.

How can the non-negativity of eigenvalues of the Laplacian be proven?

The non-negativity of eigenvalues of the Laplacian can be proven using a variety of mathematical techniques, including the method of separation of variables and the spectral theorem. The proof typically involves using the properties of the Laplacian operator and its eigenfunctions to show that all eigenvalues must be non-negative.

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