All Questions
Tagged with linear-algebra sp.spectral-theory
128 questions
2
votes
0
answers
279
views
Eigenvalues of this matrix
I have a linear map that is defined by $$T:\text{lin}(1,...,x^m) \rightarrow \text{lin}(1,...,x^m) \text{ with}$$ $$x^k \mapsto 2w(k-m)x^{k+1}+(k^2-k-w^2)x^k-2kwx^{k-1}+(k-k^2)x^{k-2}$$
Let me give a ...
- 329
1
vote
1
answer
546
views
Existence of a real eigenvalue
I have a matrix $M \in \mathbb{R}^{(n+1) \times (n+1)}$ that is tridiagonal.
In numerical computations I found out that I always find a real eigenvalue. My question is: Is there a theorem that ...
CommunityBot
- 1
2
votes
1
answer
520
views
Neighborhood overlap matrix for a bipartite graph
Let $G$ be an undirected, simple, bipartite graph with parts $V$ (having $n$ vertices) and $W$ (having $m$ vertices). Define the following $n$-by-$n$ matrix: for any $i,j \in V$, $$a_{ij} = |N_i \cap ...
CommunityBot
- 1
1
vote
0
answers
209
views
An optimization problem on the sphere
Let $S$ be a sphere centered at origin in $\Bbb R^{2n}$ of radius $\sqrt{2n}$. Let $D$ be a diagonal matrix. Let $U$ be an orthogonal matrix. Let $r\in\Bbb Z_+$ be a fixed integer.
Let vector $v\in\...
- 13.9k
29
votes
3
answers
3k
views
Perron-Frobenius "inverse eigenvalue problem"
The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
- 4,679
6
votes
1
answer
371
views
Separating the spectrum of a Hermitian matrix
Given Hermitian matrix $A$, I would like to perturbate it so that its eigenvalues become well-separated.
Specifically, let $A$ be some Hermitian matrix, and let $G$ be a Gaussian matrix, with each
...
- 445
0
votes
0
answers
117
views
"Almost orthogonalizing" matrices using a signature matrix
Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs).
Then $||AB||_{op} \leq ||A||_{op}||...
- 949
16
votes
2
answers
905
views
Eigenvalues of an "oblique diagonal" matrix
I am looking for guidance about the behavior of powers of a particular matrix (call it $A_n$ for $n\ge2$), which has come up in a counting problem about quantum knot mosaics (a good reference for ...
- 13.4k
2
votes
1
answer
358
views
positive semidefinite matrix condition
There is a great work of Alizadeh that in section 4 speaks about Minimizing sum of the first few(k-largest) eigenvalues of a symmetric matrix. Instead of a symmetric model we use the weighted ...
- 161
3
votes
1
answer
944
views
numerical range of a column-zero-sum matrix
I am trying to produce an example of a (necessarily non-normal) matrix that has only eigenvalues with positive real part, but whose numerical range contains elements with strictly negative real part. ...
- 39.9k
1
vote
1
answer
753
views
square root of a certain matrix [closed]
Hello,
I'd like to know the square root of the following $n$ by $n$ matrix, for $n > 2$ and $r>0$:
$R_{ii}=r+1$ for $i < n$
$R_{ij}=r$ otherwise
The $2$ by $2$ case is given by
$\sqrt{R}=\...
- 56.9k
2
votes
0
answers
172
views
Second eigenvalue of a weighted tree
Hello,
I am interested in upper bounding the second largest eigenvalue of the adjacency matrix of a graph $T$ with the following property:
1. $T$ contains self loops.
2. $T$ contains multiple edges (...
- 225
1
vote
1
answer
720
views
Eigenvalues of Sum of non-singular matrix and diagonal matrix
Suppose $D={\rm diag}(d_i)$ is a diagonal matrix with all diagonal entries $d_i=\pm 1$. This implies $D^2=I$.
Suppose $A$ is a non-singular Hermitian matrix. If we know that $A+A^{-1}+D$ has rational ...
- 12.1k
2
votes
0
answers
143
views
Optimization over Spectral Laplacian in cycles and trees
Is there any idea on how one can deal with an optimization problem of sum of k largest eigenvalues(min) of Laplacian matrix of a simple cycle or tree?
I would like to use semidefinite programming for ...
- 161
1
vote
0
answers
87
views
Possible diagonal values of a product of matrices with some specific characteristics
Hello all,
This is a question that might or might not be related to my previous one.
Imagine you have two matrices:
Matrix $\mathbf{\Phi}=[\Phi_1,\ldots,\Phi_M]\in\mathbb{R}^{L\times M}$ where $M\...
CommunityBot
- 1
0
votes
0
answers
146
views
Global solution for spectral clustering
I used spectral clustering for directed graphs suggested by Dengyong Zhou paper to partition the graph.I selected the eigen vectors corresponding to k largest eigen values and then I use kmeans or FCM ...
CommunityBot
- 1
3
votes
1
answer
204
views
Effects of unitarian multiplication into the spectrum of a finite matrix.
I am interested in the following problem: Let $P$ be a $n\times n$ complex finite matrix such as $PP^\dagger =W$. Given $W$, what can I say about the spectrum of $P$?
This matrix "square-root" has ...
- 188k
1
vote
1
answer
160
views
Morse index and permutation of diagonal entries of a symmetric matrix
Do there exist results concerning preservation or not of the Morse index of a symmetric matrix $A$, after permuting its diagonal entries, and keeping fixed the off--diagonal ones?
Thanks!
- 60.5k
2
votes
0
answers
291
views
Eigen-decomposition perturbation
Let $A$, $B$ and $A_k + B$ be symmetric matrices with eigenvalues $\sigma_1 \geq \sigma_2 \ldots \geq \sigma_n$, $\rho_1 \geq \rho_2 \ldots \geq \rho_n$ and $\lambda_1 \geq \lambda_2 \ldots \geq \...
- 21
8
votes
0
answers
738
views
Bounding sum of first singular values squared for Kronecker sum of traceless matrices
Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e.
$$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + B^\...
- 1,612
3
votes
1
answer
2k
views
Singular values of differences of square matrices
Suppose $A, B \in \mathbb{R}^{n \times n}$. Let $\sigma_1(A),\ldots,\sigma_n(A)$ be the singular values of $A$, and let $\sigma_1(B),\ldots,\sigma_n(B)$ be the singular values of $B$. If I know these ...
- 28.6k
9
votes
1
answer
1k
views
0 eigenvalue for a symmetric tridiagonal matrix
Let $T\in \mathbb{R}^{n\times n}$ be a symmetric tridiagonal matrix having the off--diagonal entries equal to -1. The diagonal entries are all positive, $a_i>0$, $i=\overline{1,n}$, and there ...
- 28.6k
4
votes
1
answer
314
views
Spectral Properties of $A(I-A)^{-1}$
I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - ...
- 20.2k
8
votes
2
answers
583
views
Efficiently computing a few localized eigenvectors
Let $H = \triangle + V(x) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am interested in domain decomposition for an eigenproblem involving $H$.
The lowest 1000 eigenfunctions of $H$, $ \psi_i $, can ...
CommunityBot
- 1
3
votes
3
answers
3k
views
Generalization of eigenvalues/vectors to modules?
What is the generalization of eigenvalues/vectors to modules?
To be specific, given a "vector" v in a module over some ring, and a linear "operator" O from the module to itself (please feel free to ...
- 9,132
4
votes
1
answer
1k
views
dominant eigenvector
Hi, everyone! Is there any efficient way to simplify the following tensor product
$X \otimes X + X^T \otimes X^T$, where $X$ is a square $n \times n$ matrix.
My goal is to efficiently compute the ...
- 20.2k
13
votes
2
answers
1k
views
Combinatorial proof of (a special case of) the spectral theorem?
The spectral theorem for a real $n \times n$ symmetric matrix $A$ says that $A$ is diagonalizable with all eigenvalues real. If $A$ happens to have non-negative integer entries, it can be interpreted ...
- 85.6k
1
vote
0
answers
393
views
iterated characteristic polynomials
If I have $N$ $M\times M$ symmetric positive definite matrices $A_i$ and an $N\times N$ positive semi-definite symmetric matrix B, let the $N\times N$ matrix $C_{ij}(\lambda)=B_{ij}$ for $i\ne j$ and $...
- 15.4k