# 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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### Eigenvectors & eigenvalues of "nearby" matrices

Suppose that I have a square matrix $A$ and another square matrix $B$ whose entries differ by $\varepsilon>0$. Is there any way to bound the differences in their eigenvalues and unit eigenvectors? ...
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### Show that $\lambda_j(Q'_nA_nQ_n)-\lambda_j(A_n)\to 0 \quad \text{as}\quad n\to \infty$

Let $(A_n),(Q_n)$ be sequences of $k\times k$ real matrices. Moreover suppose that $(A_n)$ is symmetric and bounded, and that $Q_nQ'_n\to I_k$ as $n\to \infty$, where convergence is with respect to ...
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### Any thought on that rank similar object?

Context : This problem arises on some exploration around Wyner's common information in information theory and the related minimization problem. Problem : Let $A$ be a $m\times n$ real matrix. Its ...
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### If a skew-symmetric real matrix has all eigenvalues zero, must it be the zero matrix?

This can be easily verified for $2\times2$ and $3\times3$ matrices, but can the result be generalised?
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### Time complexity of getting $r$-largest eigenvalues and vectors of a symmetric matrix.

$A$ is a $n \times n$ symmetric matrix. I would like to know the time complexity of calculating $r$-largest eigenvalues and vectors. When we need all eigenvalues and eigenvectors, it means $r=n$, I ...
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### Finding Matrix Corresponding to Ellipses of Cost Function

For a given cost function $$J(w)=(w-w_0)^TA(w-w_o)$$ it is known that contours of J(w) are ellipses with principal directions have angle of 45° and -45° with the horizontal axis. If eigenvalues of A ...
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### Graph Laplacian for weight matrix with negative edges

How can I normalize my weight matrix to get a positive semi-definite Laplacian, if I am using a weight matrix with negative edges?
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### An eigenfunction inequality for integral operators

I have an integral operator $T$ defined with respect to a positive semidefinite kernel function $k: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ probability measure $\mu(dx) = p(x) dx$ defined ...
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### Is the assumption that $V$ is finite-dimensional really necessary in Exercise 5.A.28? (Sheldon Axler "Linear Algebra Done Right 3rd Edition")

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. The author assumed that $V$ is finite-dimensional in Exercise 5.A.28. But I don't think that this assumption is ...
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### Singular vectors from true and sample covariance do not align

Suppose we have a covariance matrix $C\in\mathbb{R}^{p\times p}$ which is diagonal with exponential decaying values. This covariance matrix is used to sample $n$ data points $x\sim\mathcal{N}(0,C)$ ...
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### Confusing statement regarding Routh-Hurwitz criterion

I'm currently reading "On the Solutions and the Steady States of a Master Equation" by Joel Keizer. Keizer introduces a matrix $\Lambda$ with the following properties: $-\Lambda_{ij} \geq 0$ ...
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### Change of eigenvalues under near orthogonal matrix multiplication

Let $A,B\in\mathbb{R}^{d\times d}$ be positive definite diagonal matrices. Let $\Phi\in\mathbb{R}^{n\times d}$ satisfy $\text{rank}(\Phi) = n \leq d$ and be such that $\Phi\Phi^T = I_n$. When $d = n$, ...
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I have a compact self-adjoint positive integral operator $Q:L^2(0, \infty) \to L^2(0, \infty)$ with operator norm $\| Q\| =1$. By the assumptions, we know $1$ is an eigenvalue of $Q$. Let $y\ne 0$ and ...
### Determinant of circulant $(0,1)$ matrices of certain form
I am interested in computing the determinant of the following circulant matrices: let $n=p^k$ for $p$ a prime and $k\in \mathbb{N}$, take $a\in \mathbb{N}$ to be such that $a<p$ and $(a,p)=1$. ...