Questions tagged [eigenvalues]

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26 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 ...
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2answers
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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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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1answer
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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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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1answer
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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1answer
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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1answer
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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1answer
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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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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1answer
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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1answer
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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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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2answers
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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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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1answer
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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1answer
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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2answers
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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2answers
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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2answers
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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1answer
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 ...
2
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1answer
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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1answer
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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1answer
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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35 views

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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1answer
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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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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32 views

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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0answers
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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0answers
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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1answer
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 $B = U^{-1}A$, ...
3
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2answers
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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0answers
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 \...

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