## Tagged Questions

1answer
131 views

### Coercive Symmetric Bilinear form on a Hilbert space

I need to show one of the two following equivalent results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance. 1) Consider a co …
1answer
156 views

### Coercive Symmetric Bilinear form on a Hilbert space

I need to show the two following results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance. Consider a continuous symmetric bi …
0answers
70 views

### Relation between the eigenspace of a covariance matrix and eigenspace of correlation matrix

I was discussing applying Principal Component Analysis to a covariance matrix versus applying PCA to the corresponding correlation matrix with a collegue. This led me to think abou …
0answers
31 views

### is there any fast algorithm for tree graph eigendecomposition?

Is there any fast algorithm that can performs the eigendecomposition of the Laplacian matrix of a tree graph? Thank you!
0answers
50 views

### Eigenvectors of contraction times projection

Suppose $A$ is a real $n\times n$ matrix with real eigenvalues: $$1=\lambda_1>|\lambda_2|\ge \ldots\ge |\lambda_n|>0.$$ Suppose $B$ is an involution, for simplicity let us assum …
1answer
123 views

### Sum of commuting semisimple operators

Let $V$ be a finite dimensional vector space over a field $K$. An operator $T:V\to V$ is called semi-simple if every $T$-invariant subspace of $V$ has a complement(for algebraicall …
2answers
437 views

### Singular Value Decomposition of Noisy Matrices

I am an engineer who makes measurements of a variable over a grid of, say, $m\times n$. Since these are actual measurements, the true values are always corrupted by noise, and what …
2answers
289 views

### Eigenvalues of a Symmetric Positive Semi-Definite (PSD) matrix after rank one update

I have a Symmetric Positive Semi-Definite matrix $A$ which i know its eigenvalue and eigenvectors. let $v$ and $u$ be a random column vector. i want to know if it is possible to ha …
3answers
438 views

### Eigenvectors and eigenvalues of nonsymmetric Tridiagonal matrix

Hi, the question is following: We have one matrix \begin{pmatrix} -\beta & \Delta & 0 & 0 &\cdots & 0 & 0 & 0 \newline \beta & -(\beta+\Delta) …
0answers
158 views

### Comparing eigenvalues of two matrices [closed]

Suppose we have $A=\left(\begin{array}{cccc} 1 & 1 & 1 & 0\\ 1 & 3 & 0 & 0\\ 1 & 0 & 2 & 1\\ 0 & 0 & 1 & 3 \end{array}\right)$ and …
1answer
196 views

### Eigenvectors of asymmetric graphs

Let $G$ be an asymmetric connected graph. Then is it always the case that at least one of the eigenvectors of its adjacency matrix $A$ consists entirely of distinct entries? Thank …
4answers
2k views

### Eigenvectors and eigenvalues of Tridiagonal matrix

Hi, is it possible to analytically evaluate the eigenvectors and the eigenvalues of a tridiagonal matrix of the form :  \mathcal{T}^{a}_n(p,q) = \begin{pmatrix} 0 & q &a …
2answers
408 views

### Perturbation theory for the generalized eigenvalue problem

Is there a standard reference for the perturbation theory of the generalized eigenvalue problem? More specifically, I would like to get a systematic expansion for the problem …
0answers
190 views

### Common eigenvector

I have little experience with functional analysis beyond an undergraduate basic course, and I'm dealing with the following problem: let $V$ be an infinite-dimensional locally conv …
1answer
4k views

### Difference between Principal Component Analysis(PCA) and Singular Value Decomposition(SVD)?

I am confused between PCA and SVD. The wikipedia page for PCA has this line. "PCA can be done by eigenvalue decomposition of a data covariance matrix or singular value decompositi …

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