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 $\lambda x - Tx = 0$.
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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 ...
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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 $\...
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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 ...
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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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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 ...
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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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