Number associated to a linear operator from a vector space $V$ to itself: $\lambda$ is and eigenvalue of $T\colon V\to V$ if the map $x\mapsto $\lambda x-Tx$ is not injective.
6
votes
0answers
191 views
Min Max Principle and Rayleigh-Ritz-Method for eigenvalues of unbounded operators?
Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f. http://en.wikipedia.org/wiki/Min-max_theorem
Does the following generalization of that fact also hold?
Theorem: ...
6
votes
0answers
241 views
Generalized Eigenvalue Problem with one matrix having low rank
I have a specific Generalized Eigenvalue Problem (GEVP) where i am primary not interested in solving this problem but concluding from a standard EVP the spectrum of the GEVP.
The Problem
Let $A$ be ...
5
votes
0answers
58 views
Why should the eigenvalues of random matrices reflect zeta function zeroes?
As per this article.
And also : Is this a particular property of 2 dimensional objects? Could random vectors also model the "universality phenomena" -- globally random distribution of zeroes combined ...
5
votes
0answers
96 views
Reference suggestion: eigenvalues of tridiagonal matrices
I would like to ask for a reference on the problem of computing the eigenvalues/eigenvectors of tridiagonal matrices (not necessarily with constant diagonals).
I have seen authors use continued ...
5
votes
0answers
100 views
Primes approximated by eigenvalues?
Consider the matrix starting:
$$\displaystyle T(n,k) = -\begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ ...
4
votes
0answers
183 views
Recovering a Matrix knowing its eigenvectors and eigenvalues
Given the eigenvalues and eigenvectors of a matrix $R^{n\times n}$ is that possible to recover the same matrix from smaller matrices $R^{(n-1) \times (n-1)}$ where one of its eigenvalues and ...
4
votes
0answers
171 views
Continuous spectral value of right shift operator $\ell^2(\mathbb{N})$
Let $T:\ell^2(\mathbb{N} \to \ell^2(\mathbb{N})$ be the operator that sends$(x_1,x_2,x_3,...) \to (0,x_1,x_2,x_3,....)$. I want to show that $\lambda = 1$ is in the continuous spectrum.
To approach ...
4
votes
0answers
217 views
Fourier matrix - multiplicity of eigenvalues?
This question is Miscellaneous Exercise M.10 in Chapter 8 (Bilinear Forms) of Artin's Algebra. (The sentences in italics are due to me.)
The row and column indices in the $n \times n$ Fourier ...
3
votes
0answers
27 views
If the matrix is positive definite, then its similar matrix is also positive definite?
If $A$ is positive definite and $B$ is similar to $A$.
Can we say that $B$ is also positive definite?
I guess it is true since two matrices have same eigenvalues, and if $\sigma(A) > 0$, and so is ...
3
votes
0answers
111 views
show that the function satisfies condition of the lemma
Let $(f_n)_{n\geq 0}$ be an eigenfunctions of the compact integral operator
$F$, defined on $L^2([-1,1])$ by
$$
F(f)(x)=\int_{-1}^1\frac{\sin a(x-y)}{x-y}\,\psi(y)\,\mathrm{d}y, \quad \mbox{here ...
3
votes
0answers
60 views
Jacobi's Rotation has two possibilities, why do they both result in same upper triangular magnitude norm?
The Jacobi's rotation is the complex Givens rotation (unitary similarity) that results in a zero for a specified element of a matrix. If the element
is not adjacent to the diagonal, then there are ...
3
votes
0answers
114 views
Does matrix convergence in $L^p$ imply convergence of the eigenvalues in $L^p$?
Let $A_n(x)$ be a sequence of symmetric matrix functions that converges in $L^p(\Omega)$ to $A(x)$. Is it true that the eigenvalues of $A_n(x)$, or a subsequence of these, converge to the eigenvalues ...
3
votes
0answers
384 views
quick ways to »verify« determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors …
What are easy and quick ways to »verify« determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors after caculating them?
So if I calculated determinant, minimal ...
3
votes
0answers
399 views
Numerical methods to find eigenvectors with 0 eigenvalue
I'm curious if there's any numerical way of directly finding the eigenvectors with eigenvalue 0.
If I didn't have to do it directly, I would probably do it like this in pseudocode:
...
2
votes
0answers
34 views
MATLAB eig(A,B) of a positive semi-definite matrix pair (A,B) gives complex eigenvalues
A and B both are 151 by 151 psd matrices, I want to solve the generalized eigenvalue proplem
A * v = Lambda * B * v
when using MATLAB function eig()
[V,D] = eig(A,B)
I get complex numbers in both ...
2
votes
0answers
57 views
Eigen Values of A?
I just got a quick practice question here that I think should be simple but I can't find a definitive answer.
Let $A$ be a square matrix such that $A^3=A$. What can you say about the eigen values of ...
2
votes
0answers
67 views
Is there a generalization of eigenvalues/eigenvectors to $L^p$ for $p\neq2$?
Given symmetric positive definite $m\times m$ matrix $\Sigma$, $1 \le p,$ and constant $c > 0$, consider the solution to the optimization problem:
$$
\min_{v : \left\|v\right\|_p \ge c} ...
2
votes
0answers
61 views
Solving Generalized Eigenvalue Problem perturbatively
Let me formulate the problem to convey my notation.
I have a matrix $A$ which is hermitian - and is diagonalisable by a transformation
$$ U_A A\,\,U_A^{-1} = A_{diag}$$
Now the matrix is changed, ...
2
votes
0answers
45 views
positive definite matrix and double non-negative matrix with 0-1 entries
If we have a positive definite (strictly) matrix $A$ and $M$ semi-positive definite with entries $0$ or $1$ and diagonal all ones,
What are the conditions to have $\operatorname{max eigenvalue}(A ...
2
votes
0answers
72 views
Eigendecomposition of a large symmetric block tridiagonal matrix.
I was having a go at implementing the algorithm in the paper Spectral Matting which looks neat.
In it they construct a Matting Laplacian. This is a sparse N×N symmetric matrix, where each row and ...
2
votes
0answers
59 views
Positive eigenvalues in differential-algebraic equations not appearing in time-domain simulation
I am solving a system of equations derived from power system applications. It consists of index-1 differential and algebraic equations in the form:
$$\dot{x}=f(x,y) \\ 0=g(x,y)$$
To get the ...
2
votes
0answers
35 views
maxcut and the minimal eigenvalue
For an adjacency matrix $A$ that represent a graph $G=\langle V,E\rangle$, I need to show that the maxcut is bounded by:
$$
\mathrm{maxcut} \leq \frac{1}{2}|E| - \frac{|V| \lambda_{\min}(A)}{4},
$$
...
2
votes
0answers
174 views
Visualization of Singular Value decomposition of a Symmetric Matrix
The Singular Value Decomposition of a matrix A satisfies
$\mathbf A = \mathbf U \mathbf \Sigma \mathbf V^\top$
The visualization of it would look like
But when $\mathbf A$ is symmetric we can do:
...
2
votes
0answers
64 views
Spectral/ Eigen-Value solution with a linear constraint?
Is there a spectral or eigen-value solution to finding $X$ such that $Tr(CX^TMX)$ is minimum for a symmetric matrix $C$ and a p.s.d matrix $M$. Also there is a linear constraint on the minimization ...
2
votes
0answers
146 views
Positive Definite just on a Cone
Given $A \in \mathbb{R}^{n \times n}$, $C \in \mathbb{R}^{m \times n}$, and the cone $\mathcal{C}:=\{x \in \mathbb{R}^n \mid C x \geq 0\}$,
find necessary and sufficient conditions on $(A,C)$ such ...
2
votes
0answers
51 views
First Weighted Eigenvalue of the Laplacian
Let $\Omega$ be a ball centered in the origin and let $\lambda_1(\Omega)$ be the first (or lowest) eigenvalue of the Dirichlet Laplacian in $\Omega$:
$$\lambda_1 (\Omega) =\min_{u\in H_0^1 (\Omega),\ ...
2
votes
0answers
148 views
Matrices made negative semidefinite, but not simultaneously
Consider matrices $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, such that $A$ has at least $1$ strictly positive eigenvalue.
Let $X_1, X_2 \in \mathbb{R}^{n \times n}$ such that ...
2
votes
0answers
291 views
General properties of eigenvalues of a Jacobian matrix when premultiplied by a symmetric, positive definite matrix?
For a particular engineering problem that I'm working on, I have computed a Jacobian matrix $J$ and there is another matrix $M$ associated with the problem. $M$ is known to be symmetric, real-valued, ...
2
votes
0answers
83 views
How do I compute the eigenvectors of a general matrix using the QR decomposition (PLAPACK)?
I am working on an eigensolver in PLAPACK, which has built-in functionality to find the QR decomposition of a general matrix of double precision floating point numbers. I am familiar with the concept ...
2
votes
0answers
84 views
Bounds for eigenvalues, perturbation theory
Consider $-\Delta$ defined in $H^2(\Omega)\cap H_0^1(\Omega)$, $\Omega$ a smooth bounded domain of $\mathbb{R}^n$.
Let $g\in L^{\infty}(\Omega)$, $a\leq g(x)\leq b$.
Show that, if ...
2
votes
0answers
190 views
Homogeneous Fredholm Equation of Second Kind
I'm trying to show that the eigenvalues of the following integral equation
\begin{align*}
\lambda \phi(t) = \int_{-T/2}^{T/2} dx \phi(x)e^{-\Gamma|t-x|}
\end{align*}
are given by
\begin{align*}
...
2
votes
0answers
299 views
Is there any relation between the principal eigenvalue of sub matrix and the original matrix?
I am wondering whether there is any relation between principal eigenvalue of sub matrix and the original matrix.
In fact I am facing a problem which is to select $n$ rows and $n$ columns from the ...
2
votes
0answers
231 views
Differential equation, eigenvalues and eigenfunctions
How does one find all the permissible values of $b$ for $-{d\over dx}(-e^{ax}y')-ae^{ax}y=be^{ax}y$ with boundary conditions $y(0)=y(1)=0$? I assume we have a discrete set of $\{b_n\}$ where they ...
2
votes
0answers
99 views
Principal eigenvalue
How is the principal eigenvalue of elliptic differential operator defined?
Is it just a spectral radius?
1
vote
0answers
32 views
Completeness of eigenfunctions
In my computations I have obtained a sequence of eigenvalues $\lambda_k, \; k\in \mathbb{N}$ of double multiplicity. Thus, the basis for the eigenspace of $\lambda_k$ is given by $\psi_k(x) = ...
1
vote
0answers
12 views
Eigen values of L2 projection Matrix
If I have an arbitrary set of $L^2$ functions $\{\phi_i\}$, then want to find the projection onto the subspace of $L^2$ generated by the basis, i.e $span\{\phi_i\}$, I believe I just need to solve
...
1
vote
0answers
30 views
First eigenvalue of the given linear operator
I have the following question:
Let us denote $H_2^N: = \{u\in (H^2(0,1))^2: u'(0) = u'(1) = 0\}$.
Let an operator $L:H_2^N \to (L^2(0,1))^2$ be given by
$Lu = -Du'' + Cu$, where $D$ is a positive ...
1
vote
0answers
45 views
Vandermonde question
I'm studying time series analysis and in my book I came a cross with the following proof (The proof is actually the last page, but I posted as much information as possible on the problem):
I have ...
1
vote
0answers
40 views
Help find a proof : $ \lambda $ is $f$'s eigenvalue then $f|_{V_{\lambda}} $ has Jordan's basis
Could you help me find a fairly simple proof of the following theorem?
$f: V \rightarrow V, \ \ \dim V < \infty, \ \ \lambda$ is $f$'s eigenvalue $\Rightarrow \ \ f|_{V_{\lambda}}: \ V_{\lambda} ...
1
vote
0answers
18 views
Delocalization of eigenvectors in Expanding Graphs
Given an adjacency matrix A, can we say something about whether the eigenvectors corresponding to its highest(or second-highest) eigenvalues are delocalized ? By delocalization I mean that every ...
1
vote
0answers
20 views
Latent semantics and eigenvalues of the SVD
Maybe a strange question :-) trying to figure out the relation with eigenvalues and the documents/words of and LSI? since $Av = \lambda v$, does this mean that eigenvalues describe the relation ...
1
vote
0answers
23 views
finding the decomposition of Laplacian matrix with position of zero elements unchanged
I'd like to know whether it's possible to find the decomposition of a Lapalacian matrix $A$
$B^TB = A$
where $B$ has the same dimension with $A$ and the position of zero elements in $B$ is the same ...
1
vote
0answers
35 views
Eigenvalues of discretized linear integral operator
Suppose I have the following kernel operator:
$Af(x) = \int_{-1}^1 K(x-y)f(y)dy$
which is also positive and compact. Hence, it has a countable set of positive eigenvalues. Suppose those eigenvalues ...
1
vote
0answers
62 views
Solving a Sturm-Liouville differential equation variationally
This is a problem from Haim Brezis' functional analysis book (Exercise 8.41). I solved parts of it, but am stuck on some parts/want confirmation on the method. The problem is as follows:
Let $q ...
1
vote
0answers
54 views
The value interpretation of eigenvectors.
My question is may be strange but I wanna lie it any way.
The direction of an eigenvector is the most important as we normalize it. This view is right but what about the value of this eigenvector in ...
1
vote
0answers
31 views
Characterizing (or deriving) the singular values of a matrix with structure
Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$,
$$f(x,y) = e^{\imath\pi x g(y)}$$
where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\leq x < \infty$
...
1
vote
0answers
88 views
3 nonzero distinct eigenvalues, part 2
This is an attempt to generalize the answer to a previous question
Consider the $n \times n$ matrix $$A=\left[
\begin{array}{cccc}
0 & \frac{1}{n-1} & ... & \frac{1}{n-1} \\
1 & 0 ...
1
vote
0answers
91 views
Biharmonic operator
Consider the problem:
$$ \Delta^2 u = f$$
on the square domain $U=(0,1)\times(0,1)$ with boundary conditions:
$$ u(x,y)=\Delta u(x,y) = 0$$
for $(x,y) \in \partial U.$
I try to solve it with the ...
1
vote
0answers
39 views
maximize an objective function with an infinite component
Suppose I have the following maximization problem:
$\log\det(\alpha K_p)-c\alpha$ with respect to $\alpha$ with $c$ being a constant and $m$ being the dimension of $K_p$. Here, one of the eigenvalues ...
1
vote
0answers
34 views
Singular values
I would like to ask a question about singular values of matrices of the form $A^TA$. We know that by Courant minimax principle the singular values are given by (in increasing order $s_1 > ...
