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eigenvalues of matrices or operators
21
votes
Eigenvalues of symmetric tridiagonal matrices
Amongst the polynomials that can arise as characteristic polynomials of tridiagonal matrices with zero diagonal, one finds the Hermite polynomials. Schur showed that Hermite polynomials of even degree …
9
votes
Accepted
Spectrum of an adjacency matrix
Since the eigenvalues are real, and since their sum is the trace of $A$, which is zero, we see that either all eigenvalues are zero, or there are both positive and negative eigenvalues. … For more regular graphs there are bounds on the size of cliques and cocliques that involve the least eigenvalues, and these can be rearranged to get upper bounds on the least eigenvalue. …
8
votes
Connection between eigenvalues of matrix and its Laplacian.
The short summary is that eigenvalues of $A$ provide no information useful towards computing the eigenvalues of $A+D$. …
7
votes
Connectivity of weighted graph and zero Laplacian eigenvalues
Start with the unweighted case. We have $L=BB^T$ where $B$ is (what I call) the incidence matrix of an orientation of $G$. So $B$ has one 1 and one $-1$ in each column with all other entries zero. The …
7
votes
Accepted
Non symmetric matrices with real eigenvalues
$$
\begin{pmatrix}1&0\\\0&k^{-1/2}\end{pmatrix}
\begin{pmatrix}A_1&A_2\\\ kA_2^T&A_3\end{pmatrix}
\begin{pmatrix}1&0\\\0&k^{1/2}\end{pmatrix}
= \begin{pmatrix}A_1& k^{1/2}A_2\\\ k^{1/2}A_2^T&A …
7
votes
Eigenvalues of matrix sums
If M is Hermitian then H + aI is positive semidefinite when a is large enough.
So the Hermitian + positive semidefinite case is essentially the same as the Hermitian case
4
votes
Accepted
The smallest eigenvalue from an equitable partitions
Then $\pi$ is equitable and $G/\pi$ is a path, so its eigenvalues are all simple. But if $G$ is not bipartite, its least eigenvalue is not simple. …
1
vote
Accepted
eigenvalue of Laplacian matrix
Let $z$ be the vector with all entries equal to 1. If $Az=0$ and $B=A+A^T$, then
$z^TBz=0$. Choose an orthogonal basis for $\mathbb{R}^n$ with $z$ as its first vector. If $C$ is the matrix representin …