Questions tagged [eigenvector]
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304 questions
2 votes
1 answer
197 views
Do top eigenvectors maximise both Tr$(P\Sigma)$ and Tr$(P\Sigma P\Sigma)$ for orthogonal projection matrices P?
Let $P \in \mathbb{R}^{d \times d}$ be an rank $p < d$ orthogonal projection matrix given by $P = VV^T$, $V \in \mathbb{R}^{d \times p}$. Do we have that $$ \underset{P^2 = P,\; \text{rank}(P) = p}{...
- 23
0 votes
0 answers
83 views
Smallest eigenvalues comparison between two matrices: Seeking proof ideas
This problem stems from a previous problem that has already been solved. Please refer to it. According to the solution by @Alex Gavrilov, the similarity transforming matrix from ${\bf J}$ to ${\bf ...
- 75
0 votes
0 answers
142 views
Can it be proved that the specific real symmetric matrix is positive definite? (Numerically confirmed)
${\bf A} \in {\Bbb R}^{n \times n}$ is a real symmetric positive definite M-matrix, whose non-diagonal entries are non-positive. Defining a composite matrix ${\bf J}\in {\Bbb R}^{2n \times 2n}$ as $$\...
- 75
6 votes
1 answer
478 views
Numerical tests show the smallest eigenvalue of a certain matrix remains invariant when some parameters vary – any proof ideas?
I've numerically verified that the smallest eigenvalue of a specific matrix remains invariant when some parameters change. Looking for theoretical proof approaches. Appreciate any insights. ${\bf A} \...
- 75
2 votes
0 answers
101 views
Inverse of a fourth order tensor and derivative of a tensor with respect to another using spectral decomposition
I am writing a UMAT code in Abaqus to simulate hyperelastic fracture. Towards that end, I have calculated the Second Piola-Kirchhoff tensor and the material consistent jacobian tensor. However, Abaqus ...
- 45
0 votes
0 answers
90 views
Transfer of spectral data from stratified surfaces to embedded graphs
Let $\Gamma \subset X$ be a finite connected $(q+1)$-regular Ramanujan multigraph, embedded as the 0,1-skeleta of a stratified (smooth away from $\Gamma$) 2-complex $X \subset \mathbb{R}^3$. A ...
- 621
23 votes
5 answers
3k views
Are eigenspaces continuous?
There is a class of results of the form "eigenvalues are continuous in square matrices" (e.g., this MSE question and its answers). An analogous question is whether the eigenspaces of an $n \...
- 3,255
2 votes
1 answer
144 views
Choosing eigenvectors continuously for positive-semidefinite matrix function of rank one
Consider a real-analytic, rank-one, matrix-valued function $M(t)\geq 0$ of single real variable $t$. Can one choose a symmetric factorization $M(t) = z(t) z^T(t)$ (where $z(t)$ is an eigenvector with ...
- 23
0 votes
1 answer
135 views
Eigenvectors for specific eigenvalues
When reading the following paper: Ed S. Coakley, Vladimir Rokhlin, A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices, Applied and Computational ...
1 vote
1 answer
128 views
Diagonalizing a linear self-adjoint matrix in a rank-deficient non-standard inner product space
I want to solve $Ax=\lambda x$ in a rank-deficient non-standard inner product space $\mathcal{S}^r$. By "rank-deficient", I mean that although $x$ is represented by a vector of $\mathbb{R}^n$...
- 11
-2 votes
1 answer
96 views
Two SPD matrices are identical?
Two SPD matrices admits eigen-decomposition $\Sigma_p=U_p S_p U_p^{\top}$ and $\Sigma_q=U_q S_q U_q^{\top}$, where $S_p$ and $S_q$ contain ordered eigenvalues that are distinct. Let $\Sigma_v=U_q^{\...
- 21
0 votes
1 answer
161 views
Let $A$ be a SPD matrix. Suppose diagonal $(A)_{ii}$ equals to its eigenvalue $\lambda_i$. Must $A$ be a diagonal matrix?
Given a symmetric positive definite (SPD) matrix with eigendecomposition $A = U \Lambda U^{\top}$ follows $\operatorname{diag}(A)=\operatorname{diag}(\Lambda)$, is it necessary to have $U=I$ so that $...
- 21
3 votes
3 answers
398 views
Symplectic inner products of eigenvectors of complex symplectic matrix
Let $S$ be a $2N\times 2N$ complex symplectic matrix ($\operatorname{Sp}(2N,{C})$) satisfying $S^\dagger JS=J$, where $$J=\begin{pmatrix}0&I_N\\-I_N&0\end{pmatrix}.$$ The symplectic inner ...
- 79
0 votes
1 answer
226 views
Can the derivative of eigenvectors with respect to its components be taken as zero if all eigenvalues are equal?
I want to ask a couple of follow up questions to the question answered on the thread "Derivative of eigenvectors of a matrix with respect to its components". I noticed that in the accepted ...
- 45
1 vote
0 answers
104 views
Breakdown of the fourier series identity $e_n e_m = e_{n+m}$ on a perturbed torus
I would like to apologize for leaving things a bit vague. I think my question could be stated much more precisely but right now that is difficult for me to do. I nevertheless think it is an ...
- 389
1 vote
0 answers
230 views
Eigenvalues and eigenvectors of the path Laplacian
Consider the Laplacian matrix of the path graph: $$ L = \begin{bmatrix} 1 & -1 & 0 & \cdots & 0 & 0\\ -1 & 2 & -1 & \cdots & 0 & 0\\ 0 & -1 & 2 & \...
- 11
1 vote
0 answers
112 views
Inequality involving random vectors and absolute values
Let $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^d$ be finite sets. Suppose random vectors $X \in \mathbb{X}$ and $Y \in \mathbb{Y}$ are sampled according to a joint distribution $\mathbb{P}_{XY}$. ...
1 vote
1 answer
181 views
Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number. We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows, $$ a_{k,l}=\begin{cases}1 & l=k+1\\ w &...
- 4,140
3 votes
1 answer
144 views
Eigenvectors of $P^\top P$ for 0/1 matrices $P$
Let $P$ be an $m \times N$ matrix of zeros and ones (think $N \gg m$), and let $\mathbf{u} \in \mathbb{R}^N$ be a unit vector satisfying $P^\top P\mathbf{u} = \lambda^2 \mathbf{u}$ for some $\lambda &...
- 161
0 votes
0 answers
114 views
Eigenvalues of N×N correlation matrices as N tends to infinity
I want to find a 𝑁×𝑁 positive definite correlation matrix, which ensures that as 𝑁 goes to infinity, only a finite number of eigenvalues remain non-zero, while the rest eigenvalues approach zero. ...
4 votes
1 answer
409 views
Some but not all eigenvectors mutually orthogonal
Suppose an $n\times n$ matrix has real entries and has $n$ real eigenvalues and its eigenvectors span $\mathbb R^n.$ Are there any interesting conditions under which $k$ of its eigenvectors are ...
- 12.8k
2 votes
0 answers
126 views
Smallest eigenvalue of certain PD matrix decreases under sparse perturbation
Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
- 103
0 votes
2 answers
493 views
How to show the following matrix has eigenvalues $-d,-d+1,...,d$?
Consider the following $(2d+1)\times (2d+1)$ matrix: $$ A = \begin{pmatrix} 0 &\frac{2d}{2} & 0 &0 & \cdots &0 & 0 \\ \frac{1}{2} & 0 & \frac{2d-1}{2} &0& \...
- 25
2 votes
1 answer
339 views
Continuity of eigenvector of zero eigenvalue
Wonder whether anyone has an idea on showing the following or to point out that it is not true: Let $A(t) \in \Re^{n \times n}$ be differentiable over an interval $I$, and it has a zero eigenvalue for ...
- 69
2 votes
0 answers
174 views
Convergence of eigenfunctions
In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
2 votes
1 answer
257 views
Lipschitz continuity of eigenprojections
This question has the same flavor of this and this questions, but asks for something stronger. Assume that $A$ is a symmetric $n \times n$ matrix, $H$ is a $n \times n$ perturbation matrix. Moreover ...
- 39
1 vote
0 answers
62 views
Random matrices: Relation between leading eigenvector and a vector in culumn space
Let $X$ be a $n\times n$ symmetric matrix with iid zero-mean random entries on and above the diagonal. Denote by $v$ the eigenvector corresponding to the largest eigenvalue of $X$. Let $a$ be a fixed $...
- 31
1 vote
0 answers
263 views
Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart
I am trying to understand the connection between the eigenspace of the continuous operator $$ H(x,y) = \frac{1}{x+y} $$ which is nothing but the square of the Laplace operator, and its discrete ...
- 33
3 votes
1 answer
200 views
Does this matrix equation always have a solution?
Let $\{A_i, i\ge 3\}$ be the matrices whose columns represent numbers from $0$ to $2^i-1$ in binary form. For example, $A_3 = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \...
- 235
10 votes
0 answers
801 views
Eigenfunctions of the integral kernel $1/(x^2 + x'^2)$
My question seems elementary, yet I could not find the solution after working on and searching for several days... I'd like to find the eigenfunctions of a simple integral kernel: \begin{equation} \...
- 101
2 votes
1 answer
294 views
Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$
Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set $$\tag{1} \int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in ...
- 1,009
1 vote
0 answers
157 views
Formula for the kernel of an operator
Let $\mathcal H$ be a Hilbert space and let $O$ be an operator. Obviously $M=O^\dagger O$ is a semi-positive definite operator and $v\in\ker M$ if and only if $v\in\ker O$. Therefore it seems to me ...
- 521
1 vote
2 answers
565 views
Eigenvectors of a non-symmetric rank-one update of a symmetric matrix
I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+uv^\top$ where $u$ $(n\times 1)$ and $v$ $(n\times 1)$ are column vector. Also, $A=yy^\top$ with $...
- 63
1 vote
0 answers
355 views
The geometrical multiplicity of the nilpotent matrices
The following point is well-known in the literature. Theorem. Let $A$ be a non-negative matrix in $M_n(\mathbb{R})$. If $A$ is nil-potent, there is a permutation matrix $P$ such that $P^tAP$ is ...
- 4,140
1 vote
0 answers
328 views
Eigenvalues/eigenfunctions of a diffusion generator
Consider the following symmetric second order diffusion operator, defined, for $\phi \in \mathcal{C}^{2,1}_c\left(\mathbb{R}\times \mathbb{R}_+\right)$, by: $$L\phi := \lambda_1 \partial_{R_1}(R_1 \...
- 63
0 votes
0 answers
94 views
Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix
Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
- 4,140
3 votes
1 answer
99 views
Why does the normalization term disappear when computing the MLE of decomposed Gaussians
Computing the Maximum Likelihood Estimator of Gaussians in arbitrary finite Hilbert spaces seems no easy task and I must admit to lamentably fail at it. The classical theory most often relies on ...
- 31
1 vote
0 answers
314 views
Construct a permutation matrix from some eigenvectors and eigenvalues
Given $n$ orthonormal vectors $v_1, \dots, v_n \in \mathbb R^d$, where $d > n$, we can show that there are many orthogonal matrices $X$ of size $d$ such that $v_1, \dots, v_n$ are eigenvectors of $...
- 111
1 vote
1 answer
260 views
Directed graph whose adjacency matrix admits only 0 as eigenvalue
Let $G$ be a directed graph 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 edge from $P_i$ to $P_j$, ($a_{i,...
- 4,140
2 votes
1 answer
146 views
The eigenvectors of adding a particular rank one matrix to the circulant matrix
Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$. Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ ...
- 4,140
2 votes
1 answer
340 views
Are these the only first eigenfunctions on a hemisphere?
Let $\mathbb{S}^2_+$ denote the closed upper hemisphere of the unit round sphere in $\mathbb{R}^3$. It is well known that the first positive eigenvalue of the Laplacian on the closed unit sphere is $2$...
- 2,127
1 vote
0 answers
223 views
QR algorithm for eigenvalues and eigenvectors of large symmetric matrices
I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices, My initial thought was to use Householder transformation with a Wilkinson shift ...
1 vote
1 answer
174 views
Matrix transformation that always works?
Consider the matrix $$A_2:= \begin{pmatrix} a & b_1 \\ b_2 & a\end{pmatrix}.$$ Let $\sigma_2 = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$, then $$\sigma_2 A_2 \sigma_2 = \begin{...
4 votes
1 answer
3k views
Eigenvalues of a rank-one update of a symmetric matrix
I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+xx^\top$ where $x$ $(n\times 1)$ is a column vector. and also $A=yy^\top$ with $y$ a $(n-1)$ rank ...
- 63
1 vote
0 answers
390 views
Eigenvalue decomposition of normalized adjacency matrix
Let $A$ be an adjacency matrix of undirected graph $G$, where $G$ is a connected graph. The normalized adjacency matrix is defined as $\hat{A}=D^{-1/2}AD^{-1/2}$, where $D$ is degree matrix of graph $...
- 11
1 vote
0 answers
103 views
Are eigenfunctions of the Dirichlet problem for the Laplace equation uniformly bounded?
Let $Q\subset \mathbb R^n$ be a bounded domain with boundary $\partial Q\in C^\infty$ and $\varphi_1,\varphi_2,\ldots$ are eigenfunctions of the Dirichlet problem for the Laplace equation in $Q$ ...
- 2,750
0 votes
1 answer
286 views
Faulty algorithm for simultaneous diagonalization?
I found a simple algorithm for simultaneous diagonalization of two commuting matrices (Nordgren - Simultaneous Diagonalization and SVD of Commuting Matrices), which seemed to be well-founded. For ...
- 103
0 votes
0 answers
139 views
The eigenstructure of the symmetric tridiagonal matrix whose entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and $$a_{1,2}=\cdots=a_{n-1,n}=1$$
Suppose that $A=(a_{kl})_{k,l=1}^n$ is a symmetric tri-diagonal matix in $M_n(\mathbb{R})$ whose diagonal entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and $$a_{1,2}=\cdots=a_{n-1,n}=1$$ Any approach to ...
- 4,140
0 votes
0 answers
153 views
Eigenvectors of the symmetric tridiagonal matrices whose entries above the diagonal are all the same
Let us consider the real symmetric tridiagonal matrix $T=(t_{kl})$ in $M_n(\mathbb{R})$ with $$t_{1,2}=t_{2,3}=\cdots=t_{n-1,n}=1$$ How can we compute the eigenvectors of $T$?
- 4,140
1 vote
0 answers
93 views
CLT of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional random matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments: $$ \mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\...
- 284