Questions tagged [eigenvalues-eigenvectors]
Eigenvalues are numbers associated to a linear operator from a vector space $V$ to itself: $\lambda$ is an eigenvalue of $T\colon V\to V$ if the map $x\mapsto \lambda x-Tx$ is not injective. An eigenvector corresponding to $\lambda$ is a non-trivial solution to $\lambda x - Tx = 0$.
14,653 questions
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Eigenvector of GSE matrix with right properties
I encountered a weird problem when trying to study a few properties of eigenvectors of matrices sampled from the Gaussian symplectic ensemble (GSE). I have encountered this while trying to understand ...
- 310
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Question about algebraic multiplicity, geometric multiplicity and their relation with diagonalisability
My friend is currently working on some linear algebra questions. The first translated question states "Knowing that a matrix $A\in\mathbb{R}^{3\times3}$ has only one eigenvalue $\lambda=2$ and $\...
- 3
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Showing the Cayley transform sends positive definite matrices to small matrices and vice versa
Given a matrix $Z\in\Bbb R^{n\times n}$, write $Z\succ0$ to mean that $\langle v,Zv\rangle>0$ when $v\ne0$. (We may say that $Z$ is positive definite, but note that $Z$ is not required to be ...
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What are the complex fixed points of the curl operator? ($\nabla \times \vec v = \vec v$)
Let $\vec{v}=(F_x,F_y,F_z)$, where the components are functions $\mathbb C^3 \to\mathbb C$ and the subscript simply denotes the coordinate. I am curious in finding non-trivial complex $\vec{v}$ such ...
- 682
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Spectrum of a generalized path graphs (Toeplitz matrix)
I am looking for the spectrum (adjacency or Laplacian) of the graph where vertices are labelled $1,2,..,n$ and $i$ and $j$ are adjacent if $|i-j|\le d$. The adjacency matrix is a symmetric Toeplitz ...
- 1
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Existence of irrational eigenvalues of a sum of representation matrices
Let $G$ be a finite non-abelian group and $H$ be a non-normal subgroup of $G$. It can be shown that there exists a non-linear irreducible unitary representation $(\rho,V)$ of $G$ such that the matrix $...
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Division by zero in eigenvectors of a $3\times3$ real symmetric matrix
I'm trying to understand division by zero cases in the eigenvector of a three-by-three real symmetric matrix, and how to avoid them.
I have the following matrix, where every value is real:
\begin{...
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Inverse Eigenvalue Porblem of Jacobi Matrix after Rank 1 Update
I am trying to solve the inverse eigenvalue problem for the following problem.
Given are all eigenvalues $\lambda$ of $J \in \mathbb{R}^{n\times n}$ and all eigenvalues $\mu$ of $A=J+xx^\top \in \...
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Is there a simple way to derive left eigenvectors from right eigenvectors in the case of a non-linear eigenvalue problem?
First I’ll recap the normal eigenvalue problem to help explain what I’m asking. Say we have an $n\times n$ matrix $A$. Then $\det(\lambda I-A)$ is its characteristic polynomial and its zeroes are the ...
- 557
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Every $k$-dimensional subspace is $T$-invariant $\implies T= \lambda I$
I'm working through this problem in Axler's Linear Algebra Done Right (4th edition).
It says:
Suppose that V is finite-dimensional and $k \in \{1,...,\dim(V)-1\}$. Suppose $T \in L(V)
$ is such that ...
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Interpretation of an equation arising in matrix perturbation on the inner product of eigenvectors, weighted by eigengaps
I have a question about an equation that is so simple that I feel like it should have a name and be analyzed, but I can't find a reference for it, so I am hoping someone here has seen this before. I ...
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Axler's proof of the existence of eigenvalues for operators in complex vector spaces
I'm aware that perhaps this proof has been analized and discussed a lot here but there's something that isn't clear to me from what I've been reading from Axler's text. How can we assert the equality ...
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Eigenvector and eigenvalue variation when the eigenvector is pertubated?
Given a positive-definite matrix $A$ and complex column vectors $\mathbf{u}$, $\mathbf{v}$, the following relation
$A\mathbf{u}=\lambda(\mathbf{u}+\mathbf{v})$
is satisfied, where $\lambda>0$. In ...
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Using eigenvalue to estimate the accuracy of the solution of a system
I'm studying for an exam and I came into the following question regarding the use of eigenvalues to estimate a solution.
I'm having trubles understating what the request is, can somebody give me a ...
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Reopened: Does every polynomial with a Perron root has a primitive non-negative integral matrix representation?
I came across this answer which claims that not every Perron number admits a primitive non-negative integral matrix representation. This seems to contradict Lind's theorem, which states:
If $\lambda$ ...
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The properties of the matrices in $PL = \Lambda_L P$
I just stumbled across a problem with an unexpected outcome. Somehow it seems it should have been obvious but maybe I am missing something and I am unsure about the reason.
I have 2 matrices
$$P = \...
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How to use Exercise 2.1 to solve Exercise 2.5(a)? (Numerical Linear Algebra by Lloyd N. Trefethen and David Bau, III.)
I am reading Numerical Linear Algebra by Lloyd N. Trefethen and David Bau, III.
On p.16 in this book:
Exercise 2.5. Let $S\in\mathbb{C}^{m\times m}$ be skew-hermitian, i.e., $S^*=-S$.
(a) Show by ...
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Eigenvalues of self-adjoint, off-diagonal, block matrix
There are many posts about the eigenvalues of
$X_2 =
\begin{pmatrix}
0_{n_1 \times n_1} & A_{n_1 \times n_2} \\
A^*_{n_2 \times n_1} & 0_{n_2 \times n_2}
\end{pmatrix}$.
Are there any ...
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Discrepancy in inverse calculated using GHEP and HEP
Say we have a matrix $A = L + \beta^{2} M$, where $\beta$ is a real scalar. The matrices $L$ and $M$ are symmetric positive semi-definite and symmetric positive definite respectively. I am interested ...
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Uniqueness of radial solution on the perturbed cylinder
Let $T>0$, $v\in C^{2,\alpha}(\mathbb{R}/T\mathbb{Z})$ whose norm is small enough, where $\mathbb{R}/T\mathbb{Z}$ denotes the circle of perimeter $T$. Define the perturbed cylinder $C_{1+v}^T$ by
$$...
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An alternative expression for $\frac{\left[zI-A\right]^{-1}}{z- \lambda }$ [duplicate]
I am following a discrete controls theory course and one of the professor's theory slides states that if $I$ is the identity matrix and $\lambda$ is not an eigenvalue of matrix $A$, it can be shown ...
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how to construct a volume element of a coordinate system warped by a kernel function
Take a grid in an arbitrary number of dimensions. I construct a graph kernel to define the connectivity of the grid, and apply that kernel to the grid to create a weighted digraph.
I Construct the ...
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Eigenvalues of a traceless matrix
It is well known that the eigenvalues $(\lambda_i)_i$ of a traceless square $n\times n$ matrix $M$ (with no other assumptions) check :
\begin{equation}
\sum_{i=1}^n \lambda_i = 0
\end{equation}
For $2\...
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Find Hermitian matrix from another matrix, preserving the real eigenvalues
Given a square matrix $M\in\mathbb{C}^{n\times n}$, I am looking for another matrix $P$ hermitian, i.e. $P=P^{\dagger}$, which has the same eigenvalues, or at least :
\begin{equation}
\lambda\in\...
- 225
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Assessing Positive Semi-Definiteness of Covariance Matrices in Random Cylindrical Shell Generation
I am working on generating random cylindrical shells using a multivariate Gaussian distribution. To construct the covariance matrix, I am employing the following function:
$$
K \left( \theta_L, z_L \...
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Proving that a matrix and its transpose have the same characteristic polynomial [duplicate]
I'm working on a linear algebra problem about characteristic polynomials and would appreciate some guidance.
For any square matrix $A$, prove that $A$ and $A^T$ have the same
characteristic ...
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I do not understand the rewriting of the scalar helmoltz equation into an eigenvalue problem done by this researcher using finite differences
I am trying to do what is done in the following article : A "“poor man’s approach” to modelling of micro-structured optical fibres". At some point in the article they consider the following ...
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Can LP calculate maximum/minimum eigenvalue of a matrix
It's not hard to show that the maximum eigenvalue of a matrix $A\in \mathbb{S}^n$ can be calculated through the following SDP:
\begin{align*}
\max&\ Tr(AX) \\
\text{subject to}& \ Tr(X) = 1\\
...
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Efficiently updating eigenpairs when bordering a symmetric matrix
Let $A\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Consider the bordered matrix
$$
B(x, v) \;=\; \begin{bmatrix}
A & v\\[2pt]
v^{\top} & x
\end{bmatrix}\in\mathbb{R}^{(n+1)\times(n+1)}.
$$...
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Minimal eigenvalue of Gram matrix generated by orthonormal basis
Consider an orthonormal basis $(\phi_i)_{i\in\mathbb N}\subset L^2(\mathscr X)$, where $\mathscr X\subset \mathbb R^n$ is some compact Euclidean space; for simplicity, we may just take $\mathscr X=[0,...
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Riemann-Liouville fractional integral operator norm and related singular value problem
For a while now I have been interested in the following (AFAIK) open problem: what is the $ L_p [0,1]$ norm of fractional integral operator $V^s: L_p[0,1] \to L_p[0,1] $ defined as
$$ (V^s f)(x) = \...
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What is the second derivative of a matrix function defined on the eigenvalues of a diagonalizable matrix using the Daleckii-Krein theorem?
What is the second derivative of a matrix function defined on the eigenvalues of a diagonalizable matrix using the Daleckii-Krein theorem? In other words, given $f:\mathbb{R}\rightarrow\mathbb{R}$, ...
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Eigenvalues of a sum of representation matrices.
Let $G$ be a finite group and $H\le G$. Let $\lbrace{\rho_1,\rho_2,\ldots,\rho_t}\rbrace$ be the set of all inequivalent, irreducible representations of $G$. Consider the sum
$$T:=\sum_{g\in G\...
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Eigenvalues of $M=\begin{pmatrix} A & J \\ J^{\top} & B \end{pmatrix}$ with $A,B$ diagonal
Let $A := \operatorname*{diag}(a_{1},\dots,a_{n})$ and $B := \operatorname*{diag}(b_{1},\dots,b_{m})$ be diagonal real matrices. Let
$$ M := \begin{pmatrix} A & J \\ J^{\top} & B \end{pmatrix} ...
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Uniform convergence at the critical parameter value of PDE
We study the eigenvalue problem of the nonlinear Poisson equation
$$\begin{equation}
\begin{cases}
\begin{aligned}
-\Delta u &= \lambda f(u), \quad u>0
\quad &&\text{in } \Omega \\
u &...
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Do top eigenvectors maximise both Tr$(P\Sigma)$ and Tr$(P\Sigma P\Sigma)$ for orthogonal projection matrices P?
Let $P \in \mathbb{R}^{d \times d}$ be an rank $p < d$ orthogonal projection matrix given by $P = VV^T$, $V \in \mathbb{R}^{d \times p}$. Do we have that
$$
\underset{P^2 = P = P^T,\; \text{rank}(P)...
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Efficiently computing the trace of products of diagonalizable matrices
Suppose that I have two matrices $A$ and $B$ which are both symmetric: $A=A^T, B=B^T$. Moreover, I know how to diagonalize both $A$ and $B$.
Now I would like to define $T=A^{1/2}BA^{1/2}$, which is ...
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Alternative attempt to prove self-adjoint spectral theorem - can this work?
I've read the standard proof of the self-adjoint spectral theorem which uses induction, but I thought of another idea and am attempting to make it work, I'd like to know if I'm on to something or if ...
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Given a matrix of the form $M=[(A,B),(0,I)]$, show $M$ is diagonalizable iff $A$ is diagonalizable
This question in the sample final for my linear algebra class has me, for the first time in this class, stumped (and the prof hasn't released a solution set):
Suppose $M$ is a matrix of the form $\...
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Extending a pair of left- and right-eigenvectors to a full eigenbasis
When a matrix is diagonalizable, it is possible to find left and right eigenvectors $u_i$ and $v_j$ such that $u_i^T v_j = \delta_{ij}$ (Kronecker delta), $u_i^T A = \lambda_i u_i^T$, and $A v_i = \...
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Existence of real eigenvalue > 1 on the imaginary axis for a rational matrix with a RHP eigenvalue of 1
Let $A(s)$ be an $N\times N$ matrix with all its elements proper rational functions in $s$ with real coefficients, and are analytic in the closed right-half plane (RHP) $\mathrm{Re}(s)\geq0$, i.e., ...
- 565
3
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2
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Maximum eigenvalue of the hermitian part of a matrix
Consider a complex square matrix $A$ with Hermitian part $H = (A + A^\dagger)/2$. My question is whether there exists a bound on the maximum eigenvalue of $H$ in terms of the maximum eigenvalues of $A$...
- 1,099
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What are the eigenvalues of this specific block tridiagonal block matrix?
Consider a matrix of the form
$
A =
\begin{pmatrix}
2I & -M & 0 & \dots & 0 \\
-M^T & I+M^TM & -M & \dots & 0 \\
\vdots & \ddots & \ddots & \dots & \...
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I don't understand Arnoldi iteration.
I understand normal power iteration: when you multiply a vector with the matrix $A$, the component with the largest eigenvalue gets multiplied by the largest factor. So if you repeatedly multply with $...
3
votes
1
answer
100
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If $Tu = 3w$ and $Tw = 3u$ then $T$ has an eigenvalue in $\{3,−3\}$
From Linear Algebra Done Right 4th edition, pg. 141 problem 22.
Suppose $T ∈ L(V)$ and there exist nonzero vectors $u$ and $w$ in $V$ such that $Tu = 3w$ and $Tw = 3u$. Prove that $3$ or $−3$ is an ...
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2
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Norm of an integral operator over $L^{2}(-\pi ,\pi )$
Let $H=L^{2}(-\pi ,\pi )$, and let
$$[Tf](x) = \cos ^2(x)\int_{-\pi }^{\pi}\sin (2y)f(y)dy.$$
I want to find $\operatorname{Ker}T$, $\operatorname{Im}T$, the eigenvalues and the correspondent ...
- 399
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votes
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140
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How to find the spectrum of $T+T^*$ for this compact operator $T$?
The question is related to this stackexchange question
We fix a parameter $0<q<1$. Let $l^2=l^2_{\geq 1}$. We define an operator $T$ on $l^2$ by
$$
T(e_n)=q^n\sqrt{1-q^{2n}}e_{n+1}.
$$
Its ...
- 1,204
2
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0
answers
107
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The spectrum of a rank one update to a compact, self adjoint operator
Consider the Hilbert-Schmidt operator $K$ defined on $L^2([0,T])$: \begin{align}
Kf(\tau) = \int_0^T \left(\mathrm{e}^{-\frac{|\tau-s|}{a}}-\mathrm{e}^{\frac{\tau + s - 2T_0}{a}}\right)f(s)\,\mathrm{d}...
- 82
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0
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Does diagonalization of a matrix imply that the eigenvectors are unique to a given form of a matrix? [duplicate]
If you have a collection of n (nonzero and unique) eigenvectors, is there a way to find a general form of an n-by-n matrix that corresponds to them in such a way that 'rules out' alternative forms?
...
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Eigenvalues of variation of the Kac matrix
I am investiganting some regularity properties for vector fields on $SU(2)$ and I reduced it to obtaining lower bounds on the absolute value of the eigenvalues (which coincide with the singular values)...
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