Questions tagged [eigenvalues]
eigenvalues of matrices or operators
818
questions
2
votes
1
answer
156
views
Minimal Laplacian spread of a graph
Laplacian spread of a graph is the difference among the largest and the second smallest Laplacian eigenvalue of the graph. Is there any result or conjecture that discusses about the graphs having ...
2
votes
0
answers
207
views
Can you compute one eigenspace without computing them all?
Maybe the simplest non-trivial settings in which the spectrum of the Laplacian be can be computed is on the round sphere $\mathbf{S}^n$, and for products of manifolds. I want to use the two as ...
- 4,968
8
votes
0
answers
391
views
Bounding eigenvalues by taking high powers of matrices: history?
Let $A$ be real symmetric matrix. It is a well-known observation that we can bound any eigenvalue $\lambda$ of $A$ by using the fact that
$$\lambda^{2 k} \leq \textrm{Tr} A^{2 k}$$
for any $k\geq 1$. ...
- 19.3k
3
votes
0
answers
243
views
Extending Ky Fan's eigenvalues inequality to kernel operators
--Migrating from MSE since it might fit better here--
Base result
The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as:
$$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \...
- 245
2
votes
0
answers
93
views
Spectrum of a Lax Pair and conservation laws of a PDE
I would like to ask a question that I had asked a few days ago on the site math.stackexchange
and I still have not received an answer.
If we have a Lax operator, we know that the spectrum of this ...
- 93
2
votes
0
answers
79
views
Perturbed Gram matrix
Let $x_t \in \mathbb{S}^{d-1}$, $\forall t\in \mathbb{N}$ and let $e_1$ be the first canonical basis vector of $\mathbb{R}^d$, ie, $e_1 = (1,0,\cdots,0)$. Let us form a Gram Matrix
$$\sum_{t=1}^T(x_t ...
- 205
0
votes
0
answers
175
views
Eigenvalues without the axiom of choice
Without the Axiom of Choice (AC), we can find models of ZF set theory in which some vector spaces have no base, and also models in which some vector spaces have bases of different cardinalities.
The ...
- 4,353
4
votes
2
answers
492
views
Is it possible to use the Laplace Transform to calculate eigenvalues?
The relationship of Eigenvalues with Gradient Descent
Let $A$ be a symmetric (and thus diagonalizable) matrix, with diagonalization
$$A=VDV^T.$$
Let us define the quadratic function
$$f(x) = x^T A x.$$...
- 357
1
vote
1
answer
149
views
If $\Vert A-B\Vert_{op}\leq \varepsilon$ then $A^{-1}$ and $B^{-1}$ are uniformly close
Let $A,B$ are two $p\times p$ positive definite matrices such that $0<\delta_0\leq \min\{\lambda_{\min}(A), \lambda_{\min}(B)\}\leq \max\{\lambda_{\max}(A), \lambda_{\max}(B)\}\leq \delta_1$. Also ...
- 329
0
votes
0
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179
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Eigenbases without the Axiom of Choice
I understand that in ZF set theory without the Axiom of Choice (AC), it is consistent to have models in which there exist vector spaces over some (unspecified) field $k$ without a basis.
So in ...
- 4,353
1
vote
1
answer
313
views
Eigenvalue perturbation under sparse perturbations
Let $A \in \{0,1\}^{n \times n}$ be an irreducible matrix whose entries are in $\{0,1\}$, and let $\lambda_1(A)$ be the eigenvalue with the largest magnitude. By Perron–Frobenius theorem, we know that ...
- 11
8
votes
2
answers
331
views
Projecting onto space of matrices with spectral radius less than one
Consider the space
$$ S = \left\{ A \in \mathbb{R}^{n \times n} : \mathrm{SpectralRadius}(|A|) \leq 1 \right\}$$
where $|A|$ is the entry-wise absolute value. Given a matrix $M \in \mathbb{R}^{n\times ...
- 123
1
vote
1
answer
154
views
Derivative of eigenpair with respect to matrix
Suppose that $A$ is real and symmetric matrix (or tensor) of dimension $3 \times 3$, with its spectral decomposition
$$A = \sum_{i=1}^3 \lambda_i\ n_i\otimes n_i$$
where $\lambda_i$, $n_i$ and $\...
- 13
1
vote
1
answer
105
views
Characterization on smallest element in affine Sobolev subspace
Suppose we are given a sequence $\phi_k$ of traces (i.e. functions defined on boundary $\partial B_1$) such that
$$
\phi_k \rightarrow 0 \;\mbox{in $L^{\infty}(\partial B_1)$}
$$
(one can consider $C^{...
- 261
1
vote
1
answer
990
views
Prove that absolute value of eigenvalue is smaller than 1 [closed]
I want to prove that the absolute value of the eigenvalues of a matrix A are smaller than 1 for $$A=\left(\begin{array}{cc}
0 & -H_{11}^{-1} H_{12} \\
-H_{22}^{-1} H_{21} & 0
\end{array}\right)...
4
votes
1
answer
162
views
What is the distribution of eigenvalues of $A^TA$, where $A \sim N(\mu, \Sigma)$?
Let $A$ be a random matrix following multivariate normal distribution $N(\mu, \Sigma)$.
What is the distribution of the eigenvalues of $A^TA$?
A reference to the literature would be most welcome.
- 43
1
vote
1
answer
258
views
How can I obtain the eigenvalues of this matrix?
Consider the following $M \times 3$ matrix
$$\mathbf F = [\mathbf h_1, \mathbf h_2, \mathbf h_3],$$
with distinct non-zero singular values $\sigma_1 >\sigma_2 > \sigma_3$, where $\mathbf h_k$'s ...
- 31
2
votes
2
answers
191
views
An $n$ eigenvalue multiplicity
Suppose we have $A_i$, $i=1\ldots n$, $n\times n$ complex matrices linearly independent. It may be conjectured that there exist $(a_1,\ldots,a_n) \in \mathbb{C}^n$ not all zero such that $\sum_{i=1}^...
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2
votes
1
answer
181
views
How can I prove a randomly generated matrix has distinct non-zero eigenvalues?
Consider the following $M×M$ matrix
$$
\mathbf A=\sum_{k=1}^K =a_k \mathbf h_k \mathbf h_k^H,(M≥K)
$$
where $a_k$'s are real values and $h_k$'s are $M×1$ randomly generated vectors, e.g., complex ...
- 31
2
votes
1
answer
281
views
Two-level correlation function of eigenvalues for large random matrices
One can define the density of eigenvalues of a $N\times N$ Hermitian random matrix $H$ as:
\begin{equation}
\rho(\lambda)=\left \langle\frac{1}{N} \operatorname{Tr} \delta(\lambda-H)\right\rangle
\end{...
- 97
11
votes
1
answer
911
views
Imaginary eigenvalues
Consider the matrix
$$A(\mu) = \begin{pmatrix} 0 & 1& 0 & 0 \\ -1 & -i\mu & 0 & i \\ 0 & 0 & 0 & 1 \\ 0 &i & -1 & i\mu \end{pmatrix}.$$
This matrix is ...
- 624
2
votes
1
answer
209
views
Similarity of two matrices
Consider the matrix, for some $\lambda \in \mathbb R$ .
$$A=\begin{pmatrix} i \lambda & -1 & i & 0 \\ 1 & 0 & 0& 0 \\ i & 0 & - i \lambda & -1 \\ 0 & 0 & 1 ...
- 624
13
votes
3
answers
2k
views
Eigenvalue pattern
We consider a matrix
$$M_{\mu} = \begin{pmatrix} 1 & \mu & 1 & 0 \\ -\mu & 1 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 &-1 & 0 & 0 \end{pmatrix}$$
One easily ...
- 133
2
votes
0
answers
204
views
Gershgorin-type bounds for smallest eigenvalue of positive-definite matrix
I would like to know if there are known results for bounding eigenvalues of positive-definite matrices, in particular gram matrices $AA^\top$ based on easily computable functions of $A$.
Gershgorin ...
- 109
4
votes
2
answers
553
views
Eigenvalues in unit disk for a 2×2 block matrix
Crossposted from Mathematics.
Consider the $2\times 2$ matrix
\begin{align*}
Q =
\begin{bmatrix}
1 & 1 \\
0 & a
\end{bmatrix}
-
\epsilon
\begin{bmatrix}
1 & 0 \\
...
- 43
3
votes
0
answers
39
views
Connection of the singular value before and after normalization
Given a matrix $P \in \mathbb{R}^{n \times d}$, we can get $P = U \Sigma V^T$ by using SVD.
Let's say, we have another matrix $P' \in \mathbb{R}^{n \times d}$, it is the $P$ matrix with normalization ...
- 31
0
votes
0
answers
110
views
Methods to find the spectrum of an operator
Suppose we have a bounded, self-adjoint operator $T$ on a set of functions $\mathcal{F}$. What kinds of methods are there to find the spectrum of $T$?
Here is the setting I'm wondering about: consider ...
- 153
3
votes
2
answers
252
views
Proof (or reference) about $\lambda_i(A+\epsilon e_je_j^*) = \lambda_i(A) + \epsilon |v_{i,j}|^2 + O(\epsilon^2).$
I'm looking for a proof (or a reference in a textbook) about the fact that
$$
\lambda_i(A+\epsilon e_je_j^*) =_{\epsilon \to 0} \lambda_i(A) + \epsilon |v_{i,j}|^2 + O(\epsilon^2),
$$
where $A$ is a ...
- 161
6
votes
0
answers
146
views
Gap between consecutive Dirichlet eigenvalues
Suppose $\Omega \subset \mathbb R^2$ is a domain with a Lipschitz boundary and let $\{\lambda_k\}_{k=0}^n$ be the eigenvalues for the Laplacian operator on $\Omega$, that is to say
$$ -\Delta \phi_k = ...
- 4,089
1
vote
0
answers
83
views
Is there any property for the eigenvalues of an Hermitian matrix on which a well-structured binary mask has been applied?
While working on a quantum-focused article, I came accross the following problem. Let $\rho$ be a positive, semi-definite, $2^{n+m}$-Hermitian matrix with unit trace ($\rho$ is a density matrix). Let $...
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3
votes
0
answers
168
views
Eigenvalue estimates for kernel integral operator for Laplace kernel on unit-sphere in high-dimensions
Let $d$ be a large positive integer and let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $K_\gamma:S_{d-1} \times S_{d-1} \to \mathbb R$ be defined by $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...
- 6,726
4
votes
0
answers
163
views
Boundary conditions for singular Sturm-Liouville problem (boundary behavior of eigenfunctions)
I am not at all an expert in Sturm-Liouville theory, but I ended up on the following Singular Sturm Liouville problem:
\begin{equation}\label{1}
(1) \ \ \ \ \ \ \ \ \ \ \ y''(t)+\frac{\theta'(t)}{\...
- 83
3
votes
1
answer
2k
views
Eigenvalues of a block matrix with zero diagonal blocks
Suppose $A$ is a $k_1\times k_2$ matrix with real entries, $k_1<k_2$. Let $M$ be the matrix
\begin{equation}
M:=\begin{pmatrix}
0_{k_1} & A\\ A^\top & 0_{k_2}
\end{pmatrix},
\end{equation}
...
- 53
0
votes
0
answers
67
views
Fast decay of eigenvector elements
Let A be a set of similar (symmetric) matrices, sharing the same eigenvalues. I understand that their eigenvectors would be different. Let us focus on one eigenvector (e.g. corresponding to the lowest ...
- 47
4
votes
0
answers
137
views
Eigenvalues of Laplacian and eigenvalues of curvature operator
Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...
- 651
0
votes
1
answer
145
views
The relationship between the first eigenfuntions and the second eigenfuntions on sphere [closed]
Recently I considered the following question: If we give a second eigenfuntions $g$ on sphere, then can we construct a first eigenfuntions $f$ by $g$? Is there any relationship between the first ...
- 47
3
votes
1
answer
384
views
Derivative of eigenvalues of a symmetric tridiagonal matrix built via the Lanczos-Arnoldi scheme
Suppose $\mathbf{A}(\mu)$ being a symmetric positive definite matrix of dimension $n$ where its elements depend parametrically on the real parameter $\mu$.
Suppose now to build the orthonormal basis ...
- 131
3
votes
1
answer
443
views
Eigenvalues of product of unitaries
Consider $d\times d$ unitary matrices $U, \, V, \, W$ such that
$$
W=UV.
$$
Suppose that the eigenvalues of $U$ and $V$ are $(e^{i\theta_1},\cdots,e^{i\theta_d})$ and $(e^{i\phi_1},\cdots,e^{i\phi_d})$...
- 1,493
0
votes
1
answer
110
views
Is there a specific name for this optimization problem?
Let $A$ be an $n\times n$ symmetric positive definite matrix with eigenvalues and eigenvectors $\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_n>0$ and $v_1,v_2,\cdots,v_n$ respectively.
We know that the ...
- 205
1
vote
1
answer
144
views
Probability finite precision random matrix has distinct eigenvalues
copied from math stack exchange
There is a theorem which says the probability/size of a random matrix having repeated eigenvalues is 0 and this result is used in many fields. What I am wondering is, ...
- 183
2
votes
1
answer
402
views
Expansion in hypergraphs
Is there a useful concept of expansion in hypergraphs, generalizing the concept for graphs (see: expander graphs)?
Of course, expander graphs can be characterized in several qualitatively equivalent ...
- 19.3k
10
votes
2
answers
584
views
Lower eigenvectors of nonnegative matrices with zero trace
Let $A$ be an $N\times N$ nonnegative matrix with all diagonal entries equal to zero and such that there is $n_0$ such that all entries of $A^{n_0}$ are strictly positive. Let $\lambda_1,\ldots, \...
- 1,897
6
votes
0
answers
106
views
Eigenvalues of splitting scheme
In numerical analysis it is common to approximate a solution to a PDE
$$u'(t) = (A+B) u(t), \quad u(0)=u_0$$
which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
- 506
7
votes
2
answers
585
views
On a matrix problem in the field $\mathbb F_2$
Given $M$ a symmetric matrix in $\mathbb F_2^{n\times n}$ having $\mathsf{det}_\mathbb R(M)\neq0$ (non-singular in reals) and satisfying $PMP'=(M+J+I)$ or $P(M+J+I)P'=M$ where $P$ is a permutation ...
- 13.7k
1
vote
0
answers
40
views
Eigenvalue bounds of a random graph with a clique
I'm looking into this paper and having some problems proving (ii) of proposition 2.1. I don't quite understand how the lemma is proved. I also read the original paper where the lemma comes from but ...
- 11
1
vote
1
answer
102
views
A monotonicity property of eigenvalues
Let $A \in S^{n}_{+}$ be a positive semi-definite matrix and $D \in S^{n}_{+}$ a diagonal matrix with all the diagonal entries no smaller than one, i.e., $D_{ii} \geq 1$ for all $i \leq n$.
I wonder ...
- 205
1
vote
0
answers
327
views
Upper bound on the sum of the smallest non-zero eigenvalues
Let $\mathcal A := \{ A_1, A_2, \dots, A_n \} \subset \Bbb R^{d \times d}$ be a set of symmetric and positive semidefinite matrices.
For a matrix $A_k \in \mathcal A$, denote its (real) eigenvalues by ...
- 245
2
votes
0
answers
115
views
Formulas to determine the value of graph energy with addition or deletion of edges
If $G$ is a graph, then the graph energy of $G$ denoted by $E(G)$ is defined as the sum of absolute values of eigenvalues of the adjacency matrix of $G$. It is known that $E(G)\geq E(G-v)$, where $ ...
- 203
1
vote
0
answers
66
views
Conjugate gradient and the eigenvectors corresponding to the large eigenvalues [closed]
I am working on an optimization problem (for example, conjugate gradient) to solve $Ax=b$, where $A$ is a symmetric positive definite matrix. I can understand that the CG (conjugate gradient) has ...
- 11
1
vote
0
answers
75
views
Spectral theorem for symmetric real tensors
Is there a definition of eigenvalues that allows to use a spectral theorem?
Let $\mathbf{T}$ be a real fully symmetric tensor of order $3$ and size $N$. Its components can be represented as $T_{ijk}\...
- 97