Questions tagged [eigenvector]
The eigenvector tag has no usage guidance, but it has a tag wiki.
293 questions
-1
votes
0
answers
81
views
Can the higher order corrections to energy be calculated the same way in degenerate perturbation theory as with non-degenerate perturbation theory?
Consider the system given by,
$$ H|n\rangle = E|n\rangle$$
where:
$H$ is the hamiltonian.
$|n\rangle$ is the eigenstate.
$E$ is the energy of the eigenstate.
Now from non-degenerate perturbation ...
- 23
1
vote
1
answer
143
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,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,713
0
votes
1
answer
158
views
Can the derivative of eigenvectors with respect to its components be taken as zero if all eigenvalues are equal?
I want to ask a couple of follow up questions to the question answered on the thread "Derivative of eigenvectors of a matrix with respect to its components".
I noticed that in the accepted ...
- 3,791
4
votes
1
answer
2k
views
Eigenvalues of a rank-one update of a symmetric matrix
I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+xx^\top$ where $x$ $(n\times 1)$ is a column vector.
and also $A=yy^\top$ with $y$ a $(n-1)$ rank ...
- 188k
1
vote
0
answers
68
views
Breakdown of the fourier series identity $e_n e_m = e_{n+m}$ on a perturbed torus
I would like to apologize for leaving things a bit vague. I think my question could be stated much more precisely but right now that is difficult for me to do. I nevertheless think it is an ...
- 243
1
vote
0
answers
64
views
Eigenvalues and eigenvectors of the path Laplacian
Consider the Laplacian matrix of the path graph:
$$
L = \begin{bmatrix}
1 & -1 & 0 & \cdots & 0 & 0\\
-1 & 2 & -1 & \cdots & 0 & 0\\
0 & -1 & 2 & \...
- 11
1
vote
0
answers
80
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 $\mathbb{P}_{XY}$. ...
3
votes
1
answer
99
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
29
views
What is known about the distribution of eigenvectors for random positive semidefinite matrices?
Let $\{x_i\}_{i=1}^n \subset \mathbb{R}^d$ be iid random vectors drawn from probability measure $P$.
Define the random $d \times d$ real positive semidefinite matrix,
$$
S_n = \frac{1}{n} \sum_{i=1}^n ...
- 380
0
votes
0
answers
103
views
Eigenvectors of tridiagonal hermitian matrix
In my paper, I investigate the coordinates of the eigenvectors of a hollow tridiagonal hermitian matrix, which is defined as:
\begin{align*}
Q_n =
\begin{pmatrix}
0 & q_{1,2} & 0 & 0 & ...
- 1
4
votes
1
answer
314
views
Some but not all eigenvectors mutually orthogonal
Suppose an $n\times n$ matrix has real entries and has $n$ real eigenvalues and its eigenvectors span $\mathbb R^n.$ Are there any interesting conditions under which $k$ of its eigenvectors are ...
- 1,422
0
votes
0
answers
77
views
Eigenvalues of N×N correlation matrices as N tends to infinity
I want to find a 𝑁×𝑁 positive definite correlation matrix, which ensures that as 𝑁 goes to infinity, only a finite number of eigenvalues remain non-zero, while the rest eigenvalues approach zero.
...
2
votes
0
answers
86
views
Smallest eigenvalue of certain PD matrix decreases under sparse perturbation
Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
- 83
0
votes
2
answers
397
views
How to show the following matrix has eigenvalues $-d,-d+1,...,d$?
Consider the following $(2d+1)\times (2d+1)$ matrix:
$$
A = \begin{pmatrix}
0 &\frac{2d}{2} & 0 &0 & \cdots &0 & 0 \\
\frac{1}{2} & 0 & \frac{2d-1}{2} &0& \...
- 24.2k
2
votes
1
answer
278
views
Continuity of eigenvector of zero eigenvalue
Wonder whether anyone has an idea on showing the following or to point out that it is not true:
Let $A(t) \in \Re^{n \times n}$ be differentiable over an interval $I$, and it has a zero eigenvalue for ...
- 12.9k
2
votes
0
answers
143
views
Convergence of eigenfunctions
In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
- 6,377
2
votes
1
answer
142
views
Lipschitz continuity of eigenprojections
This question has the same flavor of this and this questions, but asks for something stronger.
Assume that
$A$ is a symmetric $n \times n$ matrix,
$H$ is a $n \times n$ perturbation matrix.
Moreover ...
- 24.2k
1
vote
0
answers
48
views
Random matrices: Relation between leading eigenvector and a vector in culumn space
Let $X$ be a $n\times n$ symmetric matrix with iid zero-mean random entries on and above the diagonal. Denote by $v$ the eigenvector corresponding to the largest eigenvalue of $X$. Let $a$ be a fixed $...
- 31
1
vote
0
answers
183
views
Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart
I am trying to understand the connection between the eigenspace of the continuous operator
$$
H(x,y) = \frac{1}{x+y}
$$
which is nothing but the square of the Laplace operator, and its discrete ...
- 33
3
votes
1
answer
155
views
Does this matrix equation always have a solution?
Let $\{A_i, i\ge 3\}$ be the matrices whose columns represent numbers from $0$ to $2^i-1$ in binary form. For example,
$A_3 = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \...
- 6,351
1
vote
0
answers
329
views
The geometrical multiplicity of the nilpotent matrices
The following point is well-known in the literature.
Theorem. Let $A$ be a non-negative matrix in $M_n(\mathbb{R})$. If $A$ is nil-potent, there is a permutation matrix $P$ such that $P^tAP$ is ...
- 63.9k
10
votes
0
answers
654
views
Eigenfunctions of the integral kernel $1/(x^2 + x'^2)$
My question seems elementary, yet I could not find the solution after working on and searching for several days...
I'd like to find the eigenfunctions of a simple integral kernel:
\begin{equation}
\...
- 628
2
votes
1
answer
229
views
Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$
Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set
$$\tag{1}
\int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in ...
- 7,197
1
vote
0
answers
120
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
6
votes
0
answers
188
views
Measurability of eigenvalues-eigenvectors of a positive compact operator
Let $H$ be a separable Hilbert space over $\mathbb{R}$. Let ${A} = \{a\colon H\to H\,|\,a\text{ is a positive, compact linear operator}\}$.
By the spectral theorem, given $a \in A$, there are scalars $...
- 163
1
vote
2
answers
415
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 $...
- 188k
1
vote
0
answers
190
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
3
votes
1
answer
67
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 ...
- 128k
0
votes
0
answers
74
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})$ ...
- 4,058
1
vote
0
answers
238
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
198
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,...
- 22.5k
2
votes
1
answer
112
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}$ ...
- 128k
2
votes
1
answer
220
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$...
- 839
11
votes
3
answers
9k
views
Eigenvectors of a symmetric positive definite Toeplitz matrix
I wish to efficiently compute the eigenvectors of an n x n symmetric positive definite Toeplitz matrix K. A full eigendecomposition would be even better.
Although I assumed this would be a well ...
1
vote
0
answers
179
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 ...
0
votes
0
answers
149
views
Diagonalizing a specific case of symmetric block matrix
Let's consider the following block matrix
$$ M = \begin{pmatrix}D&A^T\\A&-D\end{pmatrix},$$
where $A$ and $D$ are $n \times n$ matrices. The diagonal matrix $D$ is defined by $D_{kk} = k \...
1
vote
1
answer
136
views
Matrix transformation that always works?
Consider the matrix
$$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{...
- 128k
1
vote
0
answers
293
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 $...
1
vote
0
answers
78
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 $Q$ ...
- 2,715
4
votes
2
answers
512
views
XOR circulant matrices?
Take a function $f: Z_N\rightarrow R$. Construct an $N \times N$ matrix where the $(i,j)$th element of the matrix is $f(i-j)$, where $i-j$ is interpreted mod $Z_N$. The resulting matrices are ...
- 1,171
0
votes
1
answer
226
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 ...
- 128k
0
votes
0
answers
108
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 ...
- 91.8k
0
votes
0
answers
125
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 $T$?
- 4,058
1
vote
0
answers
66
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:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\...
- 284
1
vote
1
answer
84
views
Asymptotic property of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional data matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\top.
...
- 284
3
votes
1
answer
305
views
Are Neumann Laplacian eigenfunctions in $C(\overline{\Omega})$?
Consider that $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ (in the distributional sense) such that for some $\lambda>0$ we have that:
$$\begin{cases} \Delta u(x)=\lambda u(x), & x\in\Omega\...
- 684
2
votes
1
answer
76
views
The eigenvalues of the product $WD$ for some particular matrices
Let $D$ be a diagonal matrix in $M_{2n}(\mathbb{R})$ such that $D^2=I$ and Trace$(D)$=0
Suppose that $e_k$s are the standard vectors in $\mathbb{R}^{2n}$, that is
$$e_k=(0,\cdots 0,1,0,\cdots,0)^t$$...
- 188k
1
vote
1
answer
107
views
Convergent condition of the high-dimensional submatrix of some orthogonal matrix
Let $\mathbf{V}$ be a $p\times p$ orthogonal matrix (i.e., $\mathbf{V}\mathbf{V}^\top = \mathbf{V}^\top \mathbf{V} = \mathbf{I}$) whose columns are
$$
\mathbf{V} = \begin{bmatrix} \mathbf{v}_1 & \...
- 188k
3
votes
0
answers
115
views
Recovering the matrix when the Schur decomposition of its blocks are known
Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and
$$E=\left(\begin{array}{cc}
G & X \\
X^t & H
\end{array}\right)$$
where $G,H,X$ are $m\times m$ matrices.
Suppose that $...
- 4,058
0
votes
1
answer
75
views
The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$
Let $A$ be a real symmetric matrix in $M_{2n}(\mathbb{R})$with $A^2=I_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix.
$$...
- 188k