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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What is eigenvalue decay?
I've come across the term "eigenvalue decay" in several machine learning papers recently, but I do not know what it means. I also haven't managed to find a definition online.
We are ...
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Rank of principal submatrix of annihilator matrix
In the classic linear regression setting,
$y = X\beta+\epsilon$, where $y\in \mathbb{R}^n$, $X\in \mathbb{R}^{n\times p}$, $\beta\in\mathbb{R}^p$, and $\epsilon\sim N(0,\sigma^2I)$.
$P = X(X^\top X)^...
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Question about the Spectrum of a Matrix whose Entries contain Binomial Coefficients
For all $m,n \in \mathbb N,$ define
$$A_{m,n} = \left(\frac{1}{2} \right)^{m+n} \ \binom{m+n}{m} \ \frac{m-n}{m+n}.$$ Then, for each $N \in \mathbb N,$ define the skew-symmetric $N \times N$ matrix $\...
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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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What is the Eigen values $\lambda$ in the given system? [closed]
As I have just started to revised my maths knowledge after 8 years, I have a question.
What is the Eigen values $\lambda$ in the below given (attached image) system?
$\begin{bmatrix}4 & 1 \\ 3 &...
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No eigenvalue of a graph is larger than the maximum degree
This is exercise 1.1.24 in "Graph Theory" by Bondy and Murty:
$(a)$ No eigenvalue of a Graph $G$ has absolute value greater than $\Delta$
$(b)$ If $G$ is a connected graph and $\Delta$ is ...
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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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Can any eigenvalue have multiple eigenvectors if there is degeneracy in the spectrum?
I understand that if an n-dimensional matrix has n eigenvalues, each eigenvalue must correspond to one eigenvector.
When there is multiplicity in the roots of the characteristic polynomial of a matrix,...
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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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Inverse of a symmetric positive semi-definite matrix
I would like to compute a partial inverse of a symmetric semi-definite matrix.
I read about computing the pseudoinverse of a rectangular matrix by using SVD, however with a symmetric matrix I could ...
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A curious connection between the orthocenter of a pentagon and linear algebra
This question is a follow on of this one which has received a complete answer by @Intelligenti Pauca.
I have found a way to settle the issue of the orthocenter of a pentagon (not necessarily cyclic) ...
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Finding all eigenvalues of a bounded strictly upper triangle operator in a Hilbert Space
I've been working on this question for awhile, and am not sure how to finish the argument. It feels like I'm on the right track though. I've demonstrated my approach below:
Question: Let $E$ be a ...
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Eigenvalues of an unreduced Hessenberg Matrix
This was a problem from a homework several weeks ago. I've got a test coming up, and I'm still not sure I understand it. The professor mentioned it having something to do with the $(n-1)\times (n-1)$ ...
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