Questions tagged [eigenvector]
The eigenvector tag has no usage guidance, but it has a tag wiki.
293 questions
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Eigendecomposition of the Hadamard product of a rank one symmetric matrix and a positive definite symmetric matrix
Is it possible to say anything about the eigenvalues and eigenvectors of a matrix
$X = Y \circ xx^T$
where $Y$ is a positive definite symmetric matrix with known eigen-decomposition
$Y=U\Lambda U^T$...
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4
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0
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2k
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What is the time complexity of the largest singular value and its vectors?
Full zero-error SVD on an $m \times n$ matrix $A$ would cost $O(\min(m^2n,mn^2))$. What is the time complexity if we need only the largest singular value and its corresponding vectors? I think it is $...
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3
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220
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Eigenvalues and eigenvectors of nonsymmetric complex tridiagonal matrix
I wonder if it is possible to find analytically all eigenvalues and eigenvectors of the following $2n \times 2n$ non-symmetric complex tridiagonal matrix
$$M = i \begin{pmatrix}
0 & a & 0 &...
2
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0
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28
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Comparison of principal diagonals of two positive definite matrix
Let us consider a matrix with positive elements: ${\bf X}^{k\times2}=[X_1:\ldots:X_k]'$ with $X_i=(1,X_{1i})',\;i=1,\ldots,k$. Also consider ${\bf X}^{(-1)}$ as the Moore-Penrose inverse of ${\bf X}$ ...
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1
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437
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Eigenvector of a nonnegative matrix in closed form
Consider $n\times 1$ vector $\alpha = (\alpha_{1}, ..., \alpha_{n})$, where $0<\alpha_{i}<1$, and $\sum_{i=1}^{n}\alpha_i = 1$. Construct the $n\times n$ zero-diagonal matrix $A$ with $(i,j)$-th ...
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2
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2
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335
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Solving linear system when one eigenvalue is known
I have a huge sparse linear system $Ax = b$ where I know that an eigenvalue/eigenvector pair is $1$ and a vector of all $1$'s. Can this knowledge help me in solving the linear system at all? It seems ...
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5
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3
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Proving that a certain non-symmetric matrix has an eigenvalue with positive real part
Suppose that
$X$ is the $n \times n$ matrix of all ones
$Y$ is an arbitrary $n \times n$ matrix with zeroes on the diagonal and all other entries equal to $0$ or $1$
$0 < \delta < 1$
Let $Z = ...
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2
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0
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148
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Commutation relation and eigenvectors of infinite matrices [closed]
I'm given the Matrix $A$ and $A^T$:
$A =
\begin{bmatrix}
0 & 1 & 0 & 0 & \dots \\
0 & 0 & \sqrt{2} & 0 & \dots \\
0 & 0 & 0 & \sqrt{3} & \...
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3
votes
1
answer
656
views
Upper bounds on the condition number of the eigenvector matrix
Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a_\min,a_\max]$, with $a_\min$, $a_\max>0$.
Question: Are there any upper bounds on the condition number of the ...
- 2,712
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1
answer
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SOLVED: How to retrieve Eigenvectors from QR algorithm that applies shifts and deflation
After having googled for several days without locating a definitive answer, I will try my luck here!
I have implemented a version of the QR algorithm to calculate Eigenvalues and hopefully ...
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votes
1
answer
61
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Finding a matrix with shared eigen vectors with a given matrix [closed]
If I have a known matrix A, is there a method to find a matrix B that share all the eigen vectors of Matrix A?
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4
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1
answer
1k
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complexity of computing the singular vector corresponding to the smallest singular value
It is known that the singular value decomposition of an $m \times n$ matrix $A$ is in general of complexity of the order $m n^2$, assuming that $m \ge n$. But what if we only want to compute say the ...
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1
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835
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Do these matrices have the same null space?
Let $[\theta_1,\theta_2, \dots, \theta_N]^\mathrm{T} \, \in \mathbb{R}^N$. The angles are not all identical (on the circle), i.e. $[\theta_1,\theta_2, \dots, \theta_N] \not \equiv c [1,1,\dots, 1]^\...
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7
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Eigenvectors of a matrix with entries involving combinatorics
In the question Eigenvalues of a matrix with entries involving combinatorics No_way asked about eigenvectors of $n\times n$ matrix $M$ with entries \begin{eqnarray*}
M_{ij}=(-1)^{i+j}F(n, l, i, j),
\...
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286
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Generalized eigenvalue problem with nonnegative eigenvector constraint
Consider the following problem that is known to be non-convex but can be solved as a generalized eigenvalue problem (i.e. has a global optimum solution):
$\underset{w}{\text{maximize}}\quad w^{\top}...
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4
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Differentiability of eigenvalue and eigenvector on the non-simple case
Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=...
- 1,638
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84
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Matrices with almost constant coefficient have a simple eigenvalue
As a by-product of a general result for bounded operators of a Banach space, I have the following:
A matrix $L=(\ell_{ij})_{ij}$ that has almost constant coefficients in the sense that for some $c$,...
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1
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713
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The spherical harmonics are the EIGENVECTORS of Beltrami operator [closed]
In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ...
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3
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549
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Why $M_1 \subset M_2 \not \Rightarrow N_{M_1} (\lambda) \leq N_{M_2} (\lambda)$ for eigenvalue problem? (EDIT)
We know that for a direct problem with Dirichlet Boundary Condition (with Laplacian operator) that if two domains $M_1$ and $M_2$ are such that $M_1 \subset M_2$, then $\lambda(M_2) \leq \lambda(M_1)$,...
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How can I efficiently approximate the stationary distribution of an infinite CTMC with a sparse rate matrix?
I am looking for methods to approximate the stationary distribution of an infinite CTMC with a sparse rate matrix. Each row and column of the rate matrix has a finite number of non-zero elements. ...
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2
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1
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1k
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How to retrieve eigenvectors from shifted QR algorithm?
I understand that the key to retrieve eigenvectors in the non-shifted QR algorithm is to accumulate the transformations at each steps in the following way:
$Q = \Pi_i Q_i$
Can we accumulate the ...
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2
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1
answer
968
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Eigenvectors of symmetric positive semidefinite matrices as measurable functions
I'm currently interested in how discontinuous can get the eigenprojections of a continuous function taking values in a particular subspace of symmetric matrices.
I've been searching everywhere for an ...
2
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0
answers
463
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Conditions for continuity of non-simple eigenvectors
Here, https://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...
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1
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985
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Exact eigenvalues of a specific tridiagonal matrix
I'm studying the following tri-diagonal matrix
$$
X = \begin{pmatrix}
0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\
x_0 & 0 & x_1 & 0 &\cdots & 0 & ...
- 161
0
votes
1
answer
106
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Eigenvalue-related statements [closed]
(I understand this question might not be appropriate for this website, but it has been asked on MathStackexchange and did not receive any replies even with a bounty)
How can I prove that the ...
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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}(...
- 221
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1
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353
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Almost sure convergence of smallest eigenvector of diagonal matrix
I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily ...
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7
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1
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412
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Sum of the absolute eigenvalues of A>=B
Kindly help me to prove/disprove the following statement.
Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ (...
3
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3
answers
357
views
Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?
Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & ...
4
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2
answers
477
views
Non-asympototic version of Gelfand's formula
Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true.
There exists universal ...
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4
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0
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463
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The distribution of the elements of an eigenvector of random matrices
Suppose a random matrix $A$ with its elements following Gaussian distribution with non-zero mean. We know that the eigenvalues of $A$ have two patches: one is at the real axis that is far away from ...
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1
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0
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122
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Algorithm for finding eigenfunctions
I have an $ L^2(\mathbb{R}) $ operator that looks like
$$
\Omega = \int \partial\phi(a, b)\ \ |b, a\rangle \langle b, a |,
$$
where $ \langle x | a, b \rangle = f_a(x - b) e^{x^2/2} $ and $ f_a \in L^...
user66731
2
votes
1
answer
311
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Is there an algorithm to test whether a vector is an eigenvector of a power of a matrix?
Given a square matrix $A\in k^{n\times n}$ and a vector $x\in k^n$ over some field $k$, is there an algorithm to test whether there are $s\in\mathbb{N}$ and $\lambda\in k$ such that $A^sx=\lambda x$? ...
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0
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158
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Interpreting (Fiedler) spectral bisectioning
I would appreciate help on how to interpret the results of spectral bisectioning of a graph.
Given a $G=(V,E)$ with size $N$ represented by $Q$ its Laplacian matrix where the eigenvalues are ordered ...
- 355
4
votes
2
answers
512
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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 ...
- 3,979
2
votes
1
answer
163
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Shared maximum eigenvector
Let us consider two arbitrary Hermitian square matrices $\mathbf{A,B}$ with the same dimension. Given $\mathbf{v}$ the eigenvector associated to the maximum eigenvalue of $\mathbf{A}$:
Are there ...
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3
votes
1
answer
247
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Concentration and Correlation for Magnitudes of Gaussian Vectors
Suppose we have a large collection of standard normal random variables $a_i\in\mathbb{R}^n$. We know by standard concentration results that if we take $m \geq C\left(t/\epsilon\right)^2n$ samples, ...
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Hadamard Product and Eigendecomposition
I just found this related question in here Q1.
Given a positive definite matrix $\mathbf{A}$, consider its eigendecomposition $(\mathbf{A}\mathbf{V} = \mathbf{V}\mathbf{D})$. Consider an arbitrary ...
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1
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0
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537
views
Epsilon-net of operator norm ball around Identity
Suppose I look at the set of matrices which are invertible and satisfy
$$
\left\|A-Id\right\|_{op}<r
$$
for some $r<1$, where $Id$ is the $n\times n$ identity matrix. An $\epsilon$-net of such ...
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7
votes
1
answer
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Eigenvectors as continuous functions of matrix - diagonal perturbations
The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
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2
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1
answer
80
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Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix? [closed]
I have a real symmetric $3\times3$ matrix $\mathbf{M}(\mathbf{r}$) which depends on $\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field $e_i(\mathbf{r})$ over $\mathbb{R}^3$...
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2
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Derivative of an eigenvector with respect to his own 3x3 real symmetric matrix
$\mathbf{C}$ is a real, positive-definitive 3x3 symmetric matrix (I am thinking about the right Cauchy-Green tensor in solid mechanics). We perform eigendecomposition and get:
$$\mathbf{C} = \sum_{i=...
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3
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1
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561
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Reducing eigenvalues of symmetric PSD matrix towards 0: effect on ratios of original matrix elements?
Let $\boldsymbol{S}$ be $k \times k$ positive semi-definite real symmetric matrix with eigen decomposition $\boldsymbol{S} = \boldsymbol{X} \boldsymbol{\Lambda} \boldsymbol{X}'$ ($\boldsymbol{\Lambda}$...
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1
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Why are 1 and -1 eigenvalues of this matrix?
This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$.
First, let's define two matrices:
...
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0
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765
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Reference: Continuity of Eigenvectors [closed]
I am looking for an appropriate reference for the following fact. I already posted on math.stackexchange, but got no answer.
For each $X \in \mathbb{R}^{n \times n}_{\text{sym}}$ (symmetric matrix),...
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0
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270
views
Eigenvalue of product of self adjoint compact operators
Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
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0
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225
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Separating Two Groups of Data using Fisher's Linear Discriminant
I found an article (starting on page 8) that gives a neat method for finding the line/plane/hyperplane that maximizes the separation between two groups of data points in n-dimensions. It uses Fisher's ...
- 101
1
vote
3
answers
1k
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Simple Spectrum of Jacobi matrices
I want to call a matrix a Jacobi matrix (cause there may be different notions of Jacobi matrices) if it is a tridiagonal matrix with positive off-diagonal entries. Now, I read that the spectrum of ...
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11
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3
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9k
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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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1
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0
answers
136
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when can I say that $UV^T$ is a permutation matrix? [closed]
suppose we have two p.s.d matrices A and B: so we can diagonalize them like this:
A= $UΛU^T$ and $B=VΣV^T$
1: on what condition for $A$ and $B$ I can say that $UV^T$ is a permutation matrix?
2: how ...
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