Questions tagged [eigenvector]
The eigenvector tag has no usage guidance, but it has a tag wiki.
293 questions
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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 $T$?
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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:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\top.
...
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Recovering the matrix when the Schur decomposition of its blocks are known
Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and
$$E=\left(\begin{array}{cc}
G & X \\
X^t & H
\end{array}\right)$$
where $G,H,X$ are $m\times m$ matrices.
Suppose that $...
- 4,058
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Finding the eigenvectors of a submatrix
Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by,
$b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$.
$b_{n+k,l}=...
- 4,058
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What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^{\frac{n-1}{2}}$ when $n$ is odd?
Suppose that $n$ is odd. The eigen values/eigenvectors of the skew-circulant matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ are successfully computed in this post.
Q. What are ...
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Are there zero entries in the eigenvector corresponding to a simple eigenvalue?
For a real symmetric matrix $M$ and a simple eigenvalue $\lambda$, under which conditions the corresponding eigenvector has no zero entries? Perhaps, this is unconditional and one can provide a proof?
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CLT of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional random matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\...
- 284
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615
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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
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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
$$
\mathbf{V} = \begin{bmatrix} \mathbf{v}_1 & \...
- 284
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The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$
Let $A$ be a real symmetric matrix in $M_{2n}(\mathbb{R})$with $A^2=I_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix.
$$...
- 4,058
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Eigenvalues of the complement of a graph
Let $A$ and $\widetilde A$ be the adjacency matrices of a graph $G$ and of its complement, respectively.
Is there any relation between the eigenvalues of $A + \widetilde A$ and the eigenvalues of $A$ ...
- 1,269
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On the eigen vectors of a diagonalizable matrix
Let us consider the space $M_n(\mathbb{C})$. By a unitary matrix $U=(u_{ij})$ we mean that $U^{-1}=(\overline{u_{ji}})$.
Q. Let $U$ be a unitary matrix. I am looking for the pairs of matrices $(D,A)$ ...
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What is the supremum of 1-dim Hausdorff measure of the nodal set of Neumann eigenfunction $u$ for planar convex domain
All descriptions of this question are limited to 2-dimension for simplicity. Recently, I read some papers on the nodal set of Laplacian eigenfunctions. Denote $\Sigma=\{u(x)=0\}$ be the nodal set of a ...
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What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix?
What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix defined recursively by $H_1=(1)$ and $$ H_N=\begin{pmatrix}H_{N/2} & H_{N/2} \\ H_{N/2} & -H_{N/2}\end{pmatrix}, $$ ...
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Differentiability of Eigenvalues - Perturbation Theory
first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...
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Perturbation of Neumann Laplacian
Consider the $N \times N$ matrix
$$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\
-1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\
-\alpha &...
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When must an eigenvector have only non-negative entries?
What would be a reasonable sufficient condition on a real symmetric matrix that would force its eigenvector with largest eigenvalue (or one of its eigenvectors with maximal eigenvalue) to have only ...
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$\ell^1$-norm of eigenvectors of Erdős-Renyi Graphs
Setting. Let $G(n,p)$ denote the usual Erdős-Renyi (random) graphs. For each such graph there is an associated Laplacian matrix $L = D - A$ where $D$ collects the degrees on the diagonal and $A$ is ...
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Trace minimization for generalized eigenvalue problem
In [1], it is shown in theorem 1.2 that for symmetric $n \times n$ matrices $A$, $B$, we have
$$
\min_{Y \in Y^*} \text{tr}(Y^TAY) =
\text{tr}(X^TAX) =
\sum_{i=1}^p \lambda_i,
$$
with
$$
\text{
$X^...
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Derivative of eigenvectors of a matrix with respect to its components
Suppose that $B$ is a real, positive-definitive symmetric ($3\times3$) matrix (more accurately, $B$ is a tensor) with distinct eigenvalues, and that we can write it as
$$
B= \sum_{i=1}^3 \lambda_{i}(...
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Derivative of eigenvectors of an Hermitian matrix
In the question "Derivative of eigenvectors of a matrix with respect to its components", Liviu Nicolaescu has provided an answer valid for a real matrix. As outlined in the following, the ...
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Eigenvectors of Kronecker Product [closed]
Conjecture If $A$ and $B$ are two complex square matrices, then every eigenvector of $A\otimes B$ is of the form $x\otimes y$, where $x$ is an eigenvector of $A$ and $y$ is an eigenvector of $B$.
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Find the eigenvectors from the QR algorithm in the unsymmetric case
It is possible to find many references describing the QR Algorithm with more or less refinements to approximate the eigenvalues of a square matrix $A\in\mathbb{R}^{n\times n}$.
I implemented a version ...
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Local energy estimate in a semiclassical regime
Let us consider $h_n=(2n+1)^{-1/2}\to 0$ as $n\to \infty$ be a small parameter, which we just write as $h$ for convenience, and $u_h : \mathbb{R} \to \mathbb{R}$ be functions satisfying $Pu_h=0$ (I ...
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Calculating second derivatives of eigenvectors of a matrix with some degenerate eigenvalues
Given real symmetric matrix $\mathbf{M}$ with eigenvalues $\lambda_i$ and eigenvectors $\mathbf{v}_i$, the derivative of an eigenvector is $$\dot{\mathbf{v}}_i = \sum_{j \ne i} \frac{\mathbf{v}_j \...
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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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Condition number for matrix of eigenvectors of a diagonalizable matrix
Let $A$ be a diagonalizable matrix, i.e., $A=SDS^{−1}$. For any matrix $A$, condition number is defined as $\kappa(A)=\|A\| \|A\|^{-1}$.
For $A$ being a diagonalizable matrix, define $G_A=\{{S: S^{-1} ...
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Eigenvectors of a tensor in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$
I want to find the critical point of tensor $f=a_0b_0c_0 + a_1b_1c_1$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, and I followed this construction:
First, I take the following partial ...
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Real eigenvectors of complex matrices
Let $A$ be a nonsingular complex $(3 \times 3)$-matrix (that is, an element of $\mathrm{GL}_3(\mathbb{C})$).
Then what are some of the best-known criteria which guarantee $A$ to have real eigenvectors ...
- 4,605
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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 $\...
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Eigenvalues and eigenvectors of non-symmetric elliptic operators
We know that the operator $A=\Delta$ with domain $D(A)=\{u\in W^{2, 2}(\Omega): u=0 \ \ \text{on } \partial\Omega\}$ (say $\Omega$ is a bounded nice domain) has eigenvalues $\lambda_1>\lambda_2\ge \...
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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 ...
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Eigenvectors of graph Laplacian for spectral clustering
I have the following questions regarding the graph Laplacian for spectral clustering:
What is the intuition behind projecting the Laplacian (D-A, where D is the degree matrix and A is the affinity ...
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Continuity of eigenvectors
Let $\mathbb{C} \ni z \mapsto M(z)$ be a square matrix depending holomorphically on a parameter $z$ with the property that $\operatorname{dim}\ker(M(z)))=1$ for $z $ away from a discrete set $D \...
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Condition for two matrices to share at least one eigenvector?
Suppose that I have two matrices $A$ and $B$, and I want them to share a common eigenvector $x$. For simplicity let's just assume that the eigenvalue associated with $x$ is $1$ for both matrices, so $...
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Can I assign the term "is eigenvector" and "is eigenmatrix" of matrix $P$ in my specific (infinite-size) case?
Remark: I asked this in MSE, the question got views and votes but seemingly no one had an answer so far.
Background: I'm rereading a couple of my exploratory (surely not research-level) math-essays ...
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Eigenvalues of adjacency matrix of a k-regular graph
If $A_G$ is the adjacency matrix of a k-regular graph, let $B = J+xA_G$, where J is the matrix whose elements are all 1s and $x\in R$ is a scalar. If $\lambda_1\geq\lambda_2\geq \dots \geq \lambda_n$ ...
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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 ...
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Centrality measures in a network with negative correlations
I have a bidirectional network where the weights of edges are based on partial correlation matrix. I have both positive and negative values as weights. Now, I want to compute centrality measures as ...
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Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix
Is it possible to analytically evaluate the eigenvectors and eigenvalues of the following $n \times n$ tridiagonal matrix
$$
\mathcal{T}^{a}_n(p,q) = \begin{pmatrix}
0 & q & 0 & 0 &...
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Constructive proof of a rational version of Perron-Frobenius?
In the following, we work with vectors and matrices whose entries are rational numbers. Inequalities between such vectors are understood to be coordinatewise: e.g., two vectors $a = \left(a_1,a_2,\...
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Eigenvector of Hadamard matrix functions
Let $X\in\mathbb{R}^{n\times n}$ with SVD $X=UDV^T$. Are there known results regarding the eigenvectors of $Y=X^{\odot g}$? I am mainly interested in simple functions such as $g(z)=z^2$, i.e. $Y_{ij}=...
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"Unimodality" of the positive eigenvector of a non-negative irreducible matrix?
Consider an eigenvalue / eigenvector problem for a matrix $A$ that is known to be non-negative and irreducible (so the Perron-Frobenius theorem applies):
$$\sum_j A_{ij} x_j = \lambda x_i$$
Here $\...
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Infinite positive matrices with probability eigenvector
Let $A$ be an infinite non-negative matrix with integer entries ($a_{ij} \geq 0, \forall i,j \in \mathbb N$).
Suppose that $A$ is irreducible, aperiodic, and recurrent. So that it satisfies the ...
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Distribution of eigenvectors of random matrices and link with the components of the matrix
Let $M$ be a real symmetric matrix of size $N$ with its components $M_{ij}$ following a normal distribution centered around 0.
Let $x\in\mathbb{R}^N$ be an eigenvector of $M$ with eigenvalue $\lambda\...
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Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix?
Good morning,
I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v_1(x),\dots, v_n(...
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What are the eigenvalues and eigenvectors of a composition of two arbitrary linear transformations? [closed]
Let $S$ and $T$ be two linear transformations on $\mathbb{R}^n$. We can find the eigenvalues and eigenvectors of $S$ and $T$. I am trying to find out the relation(s) among the eigenvalues and ...
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Simultaneous diagonalization in Matlab [closed]
Crossposted from StackOverflow. The generalised diagonalization of two matrices $A$ and $B$ can be done in Matlab via
[V,D] = eig(A,B);
where the columns of $V$ ...
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vote
2
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From one eigenvector to many, in a very local graph?
Let $\Gamma$ be an undirected graph of bounded degree $d$ with $V = \{1,2,\dotsc,N\}$ as its set of vertices, and edges only between vertices that are at a distance $\leq M$ apart (where $M$ is much ...
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Eigenvectors of a matrix
Let $M$ be a square matrix of order $n\times d$. Let $\xi_{1},\dots,\xi_{n\times d}$ be an orthonormal basis of $\mathbb{R}^{n\times d}$ formed of eigenvectors of $M$. We have
$$\xi_{i}=(\lambda_1, 0,...
