All Questions
18 questions
2
votes
1
answer
133
views
Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable?
By the circulant matrix $C$ in $M_n(\mathbb{R})$, we mean that
$$C=[e_n|e_1|\cdots|e_{n-1}]$$ where $e_1,\cdots,e_n$ are the standard basis vectors in $\mathbb{R}^n$. It is well-known that
$$C=\...
- 4,058
6
votes
1
answer
515
views
Non-diagonalizability of the adjacency matrix of a directed graph
Let $G$ be a directed graph with no multiple edges or loops and let $P_i$ be its vertices. Let $A$ be the corresponding adjacency matrix of $G$, i.e. $a_{i,j}=1$ if and only if there is a directed ...
- 4,058
2
votes
0
answers
206
views
The rank of a Laplacian-type matrix
Suppose that $M$ is an integer, symmetric matrix of order $n>2$ with the positive integers $K_1,\dotsc,K_n$ on its main diagonal, and with all the off-diagonal elements equal to $0$ or $1$ so that ...
- 23k
4
votes
1
answer
2k
views
Relation of row sums to largest eigenvalue
I know that the largest eigenvalue of a graph is bounded between the minimal and maximal row sum of the matrix. If I have a $0-1$ symetric matrix (an adjacency matrix) and I know $k$ of the rows have ...
- 133
0
votes
0
answers
41
views
Orthogonality condition of symmetric matrix pencil
Let $P(\lambda)=\lambda M−L\in \mathbb{R}^{n \times n}$ be a matrix pencil with symmetric nonsingular matrix $M$ and $L$ is a weighted Laplacian matrix of a connected graph. Clearly $(0,1_n)$ is an ...
- 21
7
votes
1
answer
641
views
Lower bound on the eigenvalues of the Laplacian
I am looking for a graph for which $2 d_{i} < \mu_{i}$, for some index $i$, where $\mu_{1} \leq \mu_{2} \leq \dots\leq \mu_{n}$ are the eigenvalues of the Laplacian matrix $L(G)$ and $d_{1} \leq d_{...
- 123
1
vote
0
answers
43
views
Is $L'L_\text{in}+L_\text{in}'L$ positive semi-definite?
Assume that $A$ is the adjacency matrix of a strongly connected directed graph, that is, $A$ is non-negative and irreducible. Let $$L_\text{in}=D_\text{in}-A',\;L=D_\text{in}-A'+D_\text{out}-A$$ where ...
- 11
7
votes
1
answer
412
views
Sum of the absolute eigenvalues of A>=B
Kindly help me to prove/disprove the following statement.
Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ (...
0
votes
2
answers
588
views
Estimating the shift in the $\lambda_{\max}$ of a matrix under a diagonal perturbation
Given a matrix $A$ and a diagonal matrix $D$, how can we estimate $\lambda_{\max}(A+D) - \lambda_{\max}(A)$? Feel free to make other assumptions about the matrices that they are all symmetric and have ...
4
votes
2
answers
202
views
Integral roots of circulant matrix
When does the circulant matrix have only integral roots?
For example: all roots of the adjacency matrix of the complete graph $K_n$ are integer, which its adjacency matrix is circulant, but in case ...
1
vote
0
answers
249
views
Is there a way to simplify this apparently huge characteristic polynomial calculation?
Say I am given the $0/1$ adjacency matrix of an undirected graph. Also I am given a representation $\rho$ of some group $G$ and an orientation has been arbitrarily chosen along each edge. Let $E^{...
- 1,893
2
votes
1
answer
221
views
Are there good ways of relating a minor to the full determinant?
Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and ...
- 1,893
3
votes
1
answer
166
views
The spectral radius of a modified graph
Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the cone or the ...
- 6,962
5
votes
1
answer
706
views
What is the largest possible operator norm of a sparse (0,1)-matrix?
Inspired by this question, I was wondering about the following problem:
Consider all $n\times n$ $(0,1)$-matrices with $k$ ones. Which of these matrices has the largest operator norm? And how does ...
- 7,633
-1
votes
1
answer
492
views
Upper bound on iterations count for power iteration algorithm
I'm stuck trying to get upper bound on iterations count for power iteration algroithm for finding first eigenvalue of adjacency matrix $A$ given tolerance value. I've tried to figure something out ...
- 89
9
votes
1
answer
3k
views
Connection between eigenvalues of matrix and its Laplacian.
Hello!
There are two definitions of graph spectrum:
1) Eigenvalues of adjacency matrix $A$.
2) Eigenvalues of Laplacian of adjacency matrix ($L$).
Different sources offer different properties based ...
- 89
2
votes
2
answers
1k
views
Positive semidefinite decomposition, Laplacian eigenvalues, and the oriented incidence matrix
Suppose $A\in\mathbb{C}^{n\times n}$ is Hermitian and positive semidefinite with some decomposition $A=BB^*$, where $B=(b_{ij})\in\mathbb{C}^{n\times m}$ (not necessarily the Cholesky decomposition). ...
- 475
21
votes
5
answers
2k
views
The middle eigenvalues of an undirected graph
Let $ \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_{2n} $
be the collection of eigenvalues of an adjacency matrix of an undirected graph $G$ on $2n$ vertices. I am looking for any work or references ...
- 784