# Questions tagged [eigenvalues]

The eigenvalues tag has no usage guidance.

603
questions

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### 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 ...

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90 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 ...

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89 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 = ...

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49 views

### Eigenvalue perturbation Problem

Consider a nonnegative matrix $\mathbb{K} \in \mathbf{M}_{n}(\mathbb{R}) $ with positive diagonal entries, which is perturbed by a small nonnegative matrix $\mathbb{E} \in \mathbf{M}_{n}(\mathbb{R}) $ ...

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16 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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41 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 $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...

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96 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)}{\...

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71 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}
...

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18 views

### Approximating singular values of the resolvent matrix for a non-Hermitian matrix

I have a pretty niche question that stems from the following answer: https://mathoverflow.net/a/79129/87974
I am interested in bounding the following quantity: $b := |e_k^T R(z)^T R(z) e_k|$, where $...

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32 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 ...

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67 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 (...

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80 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 ...

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42 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 ...

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85 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 $(e^{i\phi_1},\cdots,e^{i\phi_d})$...

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72 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 ...

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33 views

### Minimum of the positive semidefinite quadratic function

Crossposted on Math SE
Given quadratic function
$$ f(x) = \sum_{i \in L^-} \frac{\lambda_i}{v_i^T v_i}(v_i^Tx + \frac{1}{2\lambda_i}v_i^T c)^2 + \sum_{i \in L^0} \frac{1}{v_i^T v_i}(v_i^Tc \cdot v_i^...

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59 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, ...

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147 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 ...

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336 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, \...

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88 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,...

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551 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 ...

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30 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 ...

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**1**answer

44 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 ...

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104 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 ...

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35 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 $ ...

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40 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 ...

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39 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}\...

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54 views

### Is it possible to reduce eigenvalues of tensors to an matrix eigenvalue problem?

Can we construct a larger matrix $M$ such that its eigenvalues are the same as the eigenvalues of a tensor $T$ of order 3?
Let $\mathbf{T}$ be a fully symmetric tensor of order $3$ and size $N$. Its ...

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72 views

### Minimizing the largest eigenvalue of matrix product

Let $A\in\mathbb{C}^{m\times n}$, $B\in\mathbb{C}^{n\times k}$, $C\in\mathbb{C}^{k\times m}$ be given complex matrices. The objective of the optimization problem is
\begin{equation}
\mathop {\arg \min ...

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105 views

### The Eigenvalue Problem: Perturbation Theory

Let $\mathbf{K}$ be a square matrix and $\rho(\mathbf{K})$
is the spectral radius of $\mathbf{K}$. Then, If $\mathbf{M}= \mathbf{K}+\delta \mathbf{A}$ for very small $\delta$, I want to prove that
$$ ...

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272 views

### Can the eigenvalues of a real symmetric tensor be complex?

Let $T$ be a fully symmetric tensor of rank $3$ and size $N$.
Using the following definition of eigenvalues, let $x\in \mathbb{C}^N$ and $\lambda\in\mathbb{C}$ such that:
\begin{equation}
\sum_{jk}^...

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200 views

### How to find bounds on the eigenvalues of a matrix?

Given this matrix
$M=\begin{bmatrix} 0 & m-1 & 2 & n-1\\
1 & m-2& 1 & n-1\\
2 & m-1 & 0 & 2(n-1)\\
1 & m-1 & 2 & 2(n-2)\end{bmatrix},$ show that if $\...

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74 views

### Eigen problem with constrained (equal) eigenvalues

Let $\Omega$ be a symmetric and positive definite matrix. From a test of hypothesis I know that some eigenvalues are likely to be equal (the test also suggests which eigenvalues). Do you have any ...

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155 views

### Find a way to apply the MLE on Fisher or Covariance matrix to make cross-correlations

I have 2 Fisher matrixes which represent information for the same variables (I mean columns/rows represent the same parameters in the 2 matrixes).
Now I would like to make the cross-correlations ...

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375 views

### Lower-bound for smallest eigenvalue of random $k \times $k matrix $C(W)$ defined by $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$

Let $k$ and $d$ be positive integers such that $d/k:=\lambda > 1$. Let $W$ be $k \times d$ random matrix with rows $w_1,\ldots,w_k \in \mathbb R^d$ drawn iid from $N(0,(1/d)I_d)$, and define the $k ...

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137 views

### Quaternions as eigenvalues of rank 3 tensors

Let us consider a matrix $M^{(a)}$ of size $N \times N$, having $N$ eigenvalues $\lambda_i \in \mathbb{C}$.
Considering a rank-3 tensor, we can informally think of it as a sequence of $N$ matrices $M^{...

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42 views

### How can I find family of non-isomorphic graphs with specific properties?

I asked this question:
How to find non-isomorphic graphs with specific orders?
Two new questions have arisen for me.
I have a graph, $G$, with $2n$ vertices. It has one connected component of order $...

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126 views

### How to find non-isomorphic graphs with specific orders?

I work on a problem in my research. I have a graph, $G$, with $2n$ vertices. It has one connected component of order $2n-1$ and an isolated vertex. $\lambda_1\geq \lambda_2\geq \ldots \geq \lambda_{2n}...

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### Spectral norm of product of fixed matrix and random semiorthogonal matrix

Suppose I have a fixed matrix $A \in \mathbb{R}^{a \times b}$ and a random matrix $B \in \mathbb{R}^{b \times c}$ with $c < b$ where $B'B = I_c$.
I am hoping to find a concentration inequality for ...

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65 views

### If there are eigenvectors with largest components $i$ resp. $j$, then is there an eigenvector with two largest components $i$ and $j$?

Let $G=(V,E)$ be a connected (finite simple) graph with vertex set $V=\{1,...,n\}$ and let $\theta_2\in\Bbb R$ be the second-largest eigenvalue of its adjacency matrix. I wonder about the following ...

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34 views

### Find coefficient such that limit is invertible matrix (interesting and in-depth)

First the question.
Let $A \in \mathbb R^{n \times d}$ where $d > n$ and $rank(A) = n$. Let $b \in \mathbb R^{n \times 1}$.
We are interested in the sequences $\{W_k\} \subset \mathbb R^{d \times d}...

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### Characterisation of Coxeter matrices with all non-real eigenvalues having absolute value 1

Let $C$ be an invertible integer matrix. Then a matrix $M$ is called Coxeter matrix (following Sato in https://www.sciencedirect.com/science/article/pii/S0024379505001709?via%3Dihub ) when $M=-C^{-1} ...

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28 views

### Eigenvalue concentration of Wishart and inverse Wishart matrices in the isotropic Gaussian case

I'm trying to find tail bounds, a la Chernoff or Hoeffding-like expressions for the spectra of Wishart and inverse Wishart matrices, specifically in the case where it is all isotropic Gaussians.
That ...

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58 views

### Finiteness of a constrained min/max

Take a bounded subset $\Omega \subset \mathbb{R}^d$ with smooth boundary (let's say a ball), take a ball of radius $r$, $B_r \subset \Omega$, take $\alpha \in ]0,1[$. How to prove that
\begin{align*}
...

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121 views

### Eigenvalues of block matrix

Given scalars $\alpha, \beta \in \mathbb{R}$, a symmetric positive definite matrix $A \in \mathbb{R}^{n\times n}$ and a flat matrix $B \in \mathbb{R}^{m\times n}$, where $m < n$, can I say ...

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598 views

### Counting eigenvalues without diagonalizing a matrix

Today's arXiv has a paper by Pierpaolo Vivo, Index of a matrix, complex logarithms, and multidimensional Fresnel integrals, which asks the question whether it is possible to calculate the number $N(\...

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83 views

### Geometric interpretation of generalized eigenvalue problem

I'm trying to shed some light on a recurrent problem I find while studying control systems. In many of the systems I work with, their stability depends on the eigenvalues of a matrix $B = U^{-1}A$, ...

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196 views

### Is the set of two-qubit absolutely separable states convex, and if so, what are its John ellipsoids?

Let us order the four nonnegative eigenvalues, summing to 1, of a (by definition, $4 \times 4$, Hermitian, nonnegative definite, trace one) "two-qubit density matrix" ($\rho$) as
\begin{...

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72 views

### Negative eigenvalue for a periodic Sturm-Liouville problem

Let $f \in C^{\infty}([0, 2\pi])$ be a smooth function and consider the following periodic Sturm-Liouville problem:
$$\begin{cases} u''(x) + f(x)u(x) = - \lambda u(x) \\
u(0) = u(2\pi) \\
u'(0) = u'(2\...

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37 views

### Expectation of random matrix (minimum positive eigenvalue)

Assume, $M_i$ are random symmetric positive semi-definite matrices and there exists $ 0 < \mu_1 \leq \mu_2 $ such that $\mu_1 \|x\|^2 \leq x^TE[M_i]x \leq \mu_2 \|x\|^2$ holds for some $0 \neq x \...