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 $\lambda x - Tx = 0$.
14,657 questions
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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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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 ...
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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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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 ...
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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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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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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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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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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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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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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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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:
$$
K \left( \theta_L, z_L \...
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I do not understand the rewriting of the scalar helmoltz equation into an eigenvalue problem done by this researcher using finite differences
I am trying to do what is done in the following article : A "“poor man’s approach” to modelling of micro-structured optical fibres". At some point in the article they consider the following ...
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Efficiently updating eigenpairs when bordering a symmetric matrix
Let $A\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Consider the bordered matrix
$$
B(x, v) \;=\; \begin{bmatrix}
A & v\\[2pt]
v^{\top} & x
\end{bmatrix}\in\mathbb{R}^{(n+1)\times(n+1)}.
$$...
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Riemann-Liouville fractional integral operator norm and related singular value problem
For a while now I have been interested in the following (AFAIK) open problem: what is the $ L_p [0,1]$ norm of fractional integral operator $V^s: L_p[0,1] \to L_p[0,1] $ defined as
$$ (V^s f)(x) = \...
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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
$$T:=\sum_{g\in G\...
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Uniform convergence at the critical parameter value of PDE
We study the eigenvalue problem of the nonlinear Poisson equation
$$\begin{equation}
\begin{cases}
\begin{aligned}
-\Delta u &= \lambda f(u), \quad u>0
\quad &&\text{in } \Omega \\
u &...
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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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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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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., ...
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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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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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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 ...
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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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Can you check if a collection of eigenvectors are unique to a matrix of some form? [closed]
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 addition, is it possible to check for some kind ...
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