2
votes
0answers
53 views

Reference for simplicity of the principal eigenvalue of the Laplacian

i'm currently searching for a proper reference or proof to see that the first eigenvalue $\lambda \in \mathbb{R}$ of \begin{equation*} - \Delta u = \lambda u \text{ in } \Omega, \\ u \in ...
2
votes
1answer
26 views

Lower bound for the spectralradius of a matrix

Any submultiplicative norm (for example the row-sum-norm) is an upper bound for the spectralradius of a matrix A. But is there a way to get a suitable LOWER bound for the spectralradius ? ...
4
votes
1answer
70 views

Left and right eigenvectors perpendicular to each other

I just read in a textbook on numerical methods that you can always have that the right eigenvectors of a matrix can be taken as orthonormal to the left eigenvectors for a diagonalisable matrix. This ...
2
votes
1answer
41 views

Generalized eigenvalue problem; why do real eigenvalues exist?

Under this text, you can see two pairs of matrices $(A,C)$. I am currently solving the generalized eigenvalue $(A-\lambda C)v=0$ for several pairs of $(A,C)$ and found out that they also have real ...
0
votes
0answers
30 views

Eigenvalues of the linear operator $T^*T$

Let $T$ be a linear operator $T: H_1 \mapsto H_2$, where $H_1$ and $H_2$ are both Hilbert Spaces. Suppose further that $T$ is bounded, but not self adjoint. Suppose I also know that for functions ...
2
votes
1answer
57 views

Establishing an inequality between the $2^{nd}$ largest eigenvalue of $A$ and a related matrix.

Let $A$ be an irreducible, aperiodic matrix with non-negative entries, with $1 \in \ker(A - I)$, $w \in \ker(A^\top - I)$, $w_i > 0$ $\forall i$. Define $W = \text{diag}(w)$. I am studying the ...
4
votes
1answer
96 views

Condition for degenerate eigenvalues for a matrix

Given a diagonalizable matrix $M$ (that is, a normal matrix), can we determine whether the matrix has degenerate eigenvalues without explicitly calculating all the eigenvalues and eigenvectors? 1) An ...
8
votes
3answers
149 views

What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
1
vote
1answer
55 views

Question about the essential spectrum of a negative difinite operator

please on an infinite dimensional Hilbert space how to difine the essential spectrum of an operator which is negative definite ??? Please help me Thank you.
2
votes
0answers
45 views

interpretation of generalized eigenvalue/vectors in spectral graph theory

Let us say I have a symmetric graph adjacency matrix A, a degree matrix D, a laplacian L (D-A). I have a generalized eigenvalue equation $Av=\lambda Lv$. Does the eigenvalue/vectors produced in this ...
2
votes
2answers
34 views

Application of the spectrum of an operator

http://en.wikipedia.org/wiki/Spectrum_of_an_operator What is the application of the spectrum of an operator
0
votes
1answer
64 views

The eigenvalues of a compact and self-adjoint operator on Hilbert space

Show that if $K$ is a compact self-adjoint operator on Hilbert space then it has either finitely many eigenvalues or a sequence of eigenvalues $\lambda_n\to 0$ as $n\to \infty$.
0
votes
1answer
69 views

Projections in spectral decomposition.

In my quantum mechanics book the spectral decomposition of operator $A$ is given as $A=\sum\limits_j\lambda_jP_j$ where $\lambda_j$ are the eigenvalues of matrix $A$ and $P_j$ is the orthogonal ...
0
votes
1answer
47 views

Perturbation of a matrix with negative eigenvalues

Let $A$ be a square matrix with all eigenvalues negative. What is the relationship between the $\lambda_\max$ of perturbed matrix $A + X$ and the norm of the perturbation $\|X\|$? PS: I know that the ...
0
votes
0answers
16 views

Spectral Relaxation and Eigendecomposition

Suppose I have the following optimization problem: $\underset{\mathbf{x}}{\mathop{\max }}\,{{\mathbf{x}}^{T}}A{{\mathbf{x}}^{T}}$ s.t $\mathbf{x}\in {{\left\{ -1,1 \right\}}^{N}}$. One possible ...
1
vote
2answers
45 views

Why these are equivalent?

Situation: operator theory, spectrum of a operator. We consider this as definition: $\lambda$ is a eigenvalue if $\lambda x=Tx$ for some $x\ne 0$ but I see someone saying this: $\lambda ...
0
votes
0answers
33 views

A domain in which the dirichlet laplacian has eigen values of all orders

I am trying to come up with an example to the following. Construct a domain $\Omega$ in all dimensions $n \in \mathbb{N}$, such that the spectrum of the Dirichlet Laplacian on such a domain (i.e., ...
2
votes
1answer
33 views

Does non-Hermitian implies at least one complex eigenvalue?

Ok so I'm studying linear algebra and we went trough the Spectral Theorem, including and proving the fact that for every Herimitian matrix, its eigenvalues have $0$ imaginary parts. I was wondering is ...
2
votes
0answers
66 views

Change of basis and spectral theorem

I've been having trouble with such a rudimentary problem. Let us define a matrix $A$: $$A = \begin{pmatrix} 3 & 0 & -i \\ 0 & 3 & 0 \\ i & 0 & 3 \end{pmatrix}$$ A is a 3 by 3 ...
0
votes
0answers
155 views

How does determinant relate to argument of matrix function?

I have a matrix function $R \mapsto J(R)$ from $\mathbb{R}$ to the set of irreducible matrices with non-negative entries. We can assume that $J(R)$ is $d \times d$, although any solutions that work ...
0
votes
1answer
225 views

$uu^T$ is the standard matrix for the orthogonal projection of $\mathbb{R}^n$ on the subspace spanned by $u$

Let $A$ be an $n\times n$ symmetric matrix, and $u$ an eigenvector of $A$. Why is it true that $\forall x\in\mathbb{R}^n$, $x^{T}uu^{T}(I-uu^T)x=0$? If I'm able to show this is true then I can show ...
1
vote
2answers
736 views

Eigenvectors of inverse complex matrix

For a non-singular matrix, its pretty straightforward to prove that $\lambda$ is eigenvalue of $A$ if and only if $\frac{1}{\lambda}$ is eigenvalue of $A^{-1}$. Let $A$ be a non-singular matrix, $x$ ...
1
vote
2answers
78 views

Upper bound on the difference between two elements of an eigenvector

Let $W$ be the non-negative, symmetric adjacency/affinity matrix for some connected graph. If $W_{ij}$ is large, then vertex $i$ and vertex $j$ have a heavily weighted edge between them. If $W_{ij} = ...
2
votes
2answers
108 views

Eigenvalues of $A^{T}A$

Let $\lambda_{i}(M)$ denote the $i$th eigenvalue of the square matrix $M$, and $T$ denote the matrix transpose. Is it true that $|\lambda_{i}(A^{T}A)|=|\lambda_{i}(A)|^{2}$ for every square matrix ...
9
votes
1answer
288 views

An approximate eigenvalue for $ T \in B(X) $.

This is a problem from Conway’s Functional Analysis: Definition An approximate eigenvalue for $ T \in B(X) $ is a scalar $ \lambda $ such that there is a sequence of unit vectors $ x_{n} \in X $ ...
3
votes
1answer
324 views

Eigenvalues integral operator - general case

Let $T$ be an integral operator on $L^2([0,1])$, such that: $$ (Tf)(x) = \int_0^1K(x,y)f(y)dy, $$ with $K(x,y): [0,1]^2 \rightarrow \mathbb{R}$ continuous and $K(x,y) = K(y,x)$, $K(x,y)\geq0$ $ ...
4
votes
0answers
402 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 ...
2
votes
1answer
195 views

Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
3
votes
1answer
123 views

Eigenvalues of Hilbert-Schmith operator

I am having trouble determining the eigenvalues and eigenvectors of the operator $Kv(x)= \int_0^1((x+t)v(t)dt$, where the kernel is $k=x+t$. I have tried to solve the equation $Kv(x)=\lambda v(x)$, ...
1
vote
1answer
91 views

Difference in sound between a string and a pipe

I am told that I can model the vibration of a guitar string of length $l >0$ by the following Sturm-Liouville equation $$ -u'' = \lambda u \: \: \text{ on } [0,l],$$ with boundary conditions ...
1
vote
1answer
127 views

How can I calculate the eigenvalue of the following matrix?

I have a matrix $A \in \mathbb{R}^{n \times n}$ such that its elements are all non-negative values. I know that for any $k$, $A^k$ has elements on the diagonal which are smaller or equal to 1. Can I ...
1
vote
1answer
65 views

Proof that these Hessian matrix identities are similar matrices

I am wondering if $Q, P$ are similar matrices where for a function $f:\mathbb{R^n}\to\mathbb{R}$ and for a diagonal matrix $D$ $Q=I-D^{-1}\nabla^2f(x)$ and $P=I-D^{-1/2}\nabla^2f(x)D^{-1/2}$. ...
1
vote
1answer
223 views

Similar matrix proof

$A$ and $B$ are similar matrices, if $B=PAP^{-1}$ holds for a square, non-singular matrix $P$. Now am wondering if $S^{-1}T$ and $S^{-1/2}TS^{-1/2}$ are similar matrices? Am looking for a proof for it ...
1
vote
1answer
166 views

spectrum of convex combination

$A,B$ are $n\times n$ Hermitian matrices. If the eigenvalues of $A$ and $B$ are all in an interval $I$, then the eigenvalues of any convex combination of $A,B$ are also in $I$. In the book Bhatia, ...
3
votes
1answer
857 views

are there any bounds on the eigenvalues of products of positive-semidefinite matrices?

I have real positive semidefinite matrices (symmetric) $A$ and $B$, both are $n \times n$. I am looking for upper bounds and lower bounds on the $m$-th largest eigenvalue of $AB$, in terms of the ...
4
votes
0answers
280 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 ...
16
votes
3answers
964 views

Ratio of largest eigenvalue to sum of eigenvalues — where to read about it?

Let $E_j$ be the $j$th largest-magnitude eigenvalue of a real symmetric $N \times N$ matrix $M$. I've found that the ratio $$\frac{|E_1|}{\sum_{j=1}^N{|E_j|}},$$ is a measure of the "rank-one-ness" ...
18
votes
5answers
16k views

What is the difference between Singular Value and Eigenvalue?

I am trying to prove some statements about singular value decomposition, but I am not sure what the difference between singular value and eigenvalue is. Is singular value just another name for ...
2
votes
0answers
99 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 ...
1
vote
2answers
426 views

Basic Spectral Theory Problem: Finding the Point/Continuous Spectrum of an Operator

I have the following problem: Determine the point spectrum and the continuous spectrum of the operator $$(A\psi )(x)=\theta (x)(\cos x)\psi (x)$$ on $L_2(\mathbb R,dx)$, where $\theta(x)=0$ for ...