Questions tagged [eigenvalues-eigenvectors]

Question 1

Consider the $3\times3$ left-circulant matrix with first row $(0,1,1)$: $$ C = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix} $$ where each row is a left ...

Question 2

Given a sample matrix $\mathbf{X} \in \mathbb{C}^{p\times n}$, whose entry follow i.i.d bernoulli distribution (The probability of the value being either $+1$ or $-1$ is $\frac{1}{2}$). When $p$ is ...

Question 3

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 ...

Question 4

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 $\...

Question 5

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 ...

Question 6

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 ...

Question 7

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 ...

Question 8

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 $...

Question 9

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{...

Question 10

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 \...

Question 11

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 ...

Question 12

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 ...

Question 13

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 ...

Question 14

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 ...

Question 15

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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Questions tagged [eigenvalues-eigenvectors]

Eigenvalues of real left-circulant matrices

Distribution of the largest eigenvalue of a sample covariance matrix (Bernoulli distribution sample)

Eigenvector of GSE matrix with right properties

Question about algebraic multiplicity, geometric multiplicity and their relation with diagonalisability

Showing the Cayley transform sends positive definite matrices to small matrices and vice versa

What are the complex fixed points of the curl operator? ($\nabla \times \vec v = \vec v$)

Spectrum of a generalized path graphs (Toeplitz matrix)

Existence of irrational eigenvalues of a sum of representation matrices

Division by zero in eigenvectors of a $3\times3$ real symmetric matrix

Inverse Eigenvalue Porblem of Jacobi Matrix after Rank 1 Update

Is there a simple way to derive left eigenvectors from right eigenvectors in the case of a non-linear eigenvalue problem?

Every $k$-dimensional subspace is $T$-invariant $\implies T= \lambda I$

Interpretation of an equation arising in matrix perturbation on the inner product of eigenvectors, weighted by eigengaps

Axler's proof of the existence of eigenvalues for operators in complex vector spaces

Eigenvector and eigenvalue variation when the eigenvector is pertubated?

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