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Is there any relationship between eigenvalues(or spectrum) of graph Laplacian matrix and the eigenvalues of the product of a real symmetric matrix and the Laplacian matrix?

My problem at hand is as follows :

Let A=L*B.

What is the relationship between spectrum (or eigenvalues) of L with the spectrum of A?

L is Laplacian of an undirected graph, hence real symmetric and singular. B is a real symmetric matrix.

I want to show that if I increase the magnitude of eigenvalues of L, the eigenvalues of A will also increase. However, all I could find was a trace inequality relationship, and inequality doesn't necessarily lead to any conclusion.

Abhiram V P
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  • If you count geometric multiplicities, eigenvalues may disappear. Does that count as "increasing"? For example if $$ L_1 = \begin{bmatrix}0&0\0&0\end{bmatrix} \qquad L_2 = \begin{bmatrix}1&-1\-1&1\end{bmatrix} \qquad B=\begin{bmatrix}-1 &0 \ 0 & 1\end{bmatrix}$$ then going from $L_1$ to $L_2$ changes the eigenvalues from $0$ and $0$ to $0$ and $2$, certainly as good an increase as we can hope for given that the $L$s are always singular. But the eigenvalues of $L_1B$ are $0$ and $0$ whereas $L_2B$ only has a single $0$. Is that an increase? – hmakholm left over Monica Mar 26 '19 at 11:17
  • The topmost question in the "Related" list looks pretty relevant, though it is more picky about the choice for $B$. – hmakholm left over Monica Mar 26 '19 at 11:26
  • @HenningMakholm Thank you for the comment. Regarding your first question, I could not understand the difference between the two in "eigenvalues of L1B are 0 and 0 whereas L2B only has a single 0". And were you trying to give a counterexample for the statement? If B is an identity matrix, then L2B would have eigenvalues 0 and 2. I am looking for an analytical proof which says that by increasing eigenvalue of one matrix, the eigenvalue of the product is also increased. I am getting that trend in my case while using the values, but an analytical proof is what I am after. – Abhiram V P Mar 26 '19 at 12:38
  • I am asking you whether this counts as a counterexample for you, since $L_1B$ has the eigenvalue $0$ with geometric multiplicity $2$ and $L_2B$ has the eigenvalue $0$ with geometric multiplicity $1$ and no other eigenvalues. One of the eigenvalues of $L_1B$ goes from being $0$ to not being there at all! Do you consider that to be an "increase" or not? – hmakholm left over Monica Mar 26 '19 at 12:41
  • (And I don't understand why you say "If $B$ is an identity matrix", since I have explicitly specified what $B$ in my example is -- and that is not the identity matrix"). – hmakholm left over Monica Mar 26 '19 at 12:44
  • @HenningMakholm I was trying to say what I mean by 'increase' in eigenvalue. If B is identity in your example then it shows an increase in eigenvalues of A matrix. In your example, both L1B and L2B will give eigen values at 0, right ? You were saying that L1b has 0 and 0 while L2B has a single 0. – Abhiram V P Mar 26 '19 at 12:48
  • $B$ is not the identity in my example, so pretenting that my example is a different example than the one I'm giving does not tell me anything about what you think about the example I'm actually giving. And I have no idea whatsoever what you mean by "the reference question". – hmakholm left over Monica Mar 26 '19 at 12:55
  • @HenningMakholm 1)"The topmost question in the "Related" list looks pretty relevant".could you give link or title? 2) Forget about my example, now in your example, the eigenvalues of L2B and L1B are 0 and 0, right? What do you mean when you say "L2B only has a single 0", in the original comment. – Abhiram V P Mar 26 '19 at 13:01
  • The words "topmost question" are linked. – hmakholm left over Monica Mar 26 '19 at 13:08
  • The matrix $L_1B$ has the eigenvalue $0$ WITH GEOMETRIC MULTIPLICITY 2. The matrix $L_2B$ has the eigenvalue $0$ WITH GEOMETRIC MULTIPLICITY 1 and no other eigenvalues. Do you consider the $0$ eigenvalue that disappears between $L_1B$ and $L_2B$ to have "increased"? – hmakholm left over Monica Mar 26 '19 at 13:10
  • @HenningMakholm Sorry. I am not familiar with the concept multiplicity of eigenvalues. Let me read about that first. – Abhiram V P Mar 27 '19 at 08:10
  • @HenningMakholm Regarding the geometric multiplicity being reduced, no that is not a point of concern in my case. I am looking for a proof where the magnitude of the eigenvalue of the product of matrices increasing with an increase in the magnitude of the eigenvalue of one of the matrices. – Abhiram V P Mar 27 '19 at 08:25

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