# Questions tagged [eigenvalues]

For questions involving calculation or interpretation of eigenvalues or eigenvectors.

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### Appropriate negative eigenvalue correction for PCoA of genetic distances

I am trying to find the best way to represent genetic distances in a plane so that they may use them as response variables in canonical redundancy analysis (using ...
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### Why do PCA and PCoA give the same components but different explained variances?

I'm quite familiar with Principal Component Analysisis, as I use it to study genetic structure. Lately, I was revisiting some of the functions I was using in R (...
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### Can I use combination of eigenvectors as a single vector to explain most of variance?

I have a problem trying to find a combination (or weighted average) of variables (statistics) that best explains the sample statistics. A – n x p matrix (n: observations p: variables, here are ...
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### Interpretation of Square of Covariance Matrix

I have a random variable, distributed as a sum of independent chi-squared random variables each with one degree of freedom. $$X = \sum_{i=1}^n \lambda_i \chi^2_{i(1)}$$ where $\lambda_i$ are ...
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### Data compression using either Singular Values or Eigenvalues

In many applications, an SVD of a matrix is used to determine which features are important and which ones less important. For example, in image compression, the smallest singular values are often ...
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### How do I apply AIC to test out how many eigenvalues are different from 1 in a diagonalised covariance matrix?

Given that we have $x_1$,...,$x_n$ with dimensions $p * 1$ each, and $X_i$ ~ N(0,Σ), we form the diagonalised covariance matrix Σ such that the first K eigenvalues are unknown and the ...
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### Connection between canonical correlation and distribution of roots of characteristic equation

I'm trying to make sense of the following sentence from introduction "Multiple discoveries: Distribution of roots of determinantal equations" http://statweb.stanford.edu/~ckirby/ted/papers/...
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### Intuition of KernelPCA

I'm dealing currently with kernels and kernel PCA. For this purpose I've been reading a few papers on these topics. In this context I've been reading the paper "Kernel Principal Component Analysis" by ...
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### Which of the following is NOT true regarding eigenvalues?

I am having a hard time trying to figure out the correct answer to this question. Any insight? I don't understand this at all. Which of the following is NOT true regarding eigenvalues? Option 1: An ...
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### Projection on weighted kernel PCA basis

I'm performing a sort of weighted kernel PCA, where the weights of samples can be negative. The weights of all samples are given by the diagonal weight matrix $D$. The data matrix is the $n \times d$ ...
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### The effect of non-positive-definite covariance matrix (in $p>n$ case) on PCA

Gene data has large number of dimensions as compared to samples. This leads to a non-positive-definite covariance matrix. In R when I try to use princomp which does ...
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### Different order and signs of eigenvectors when doing PCA via eig() or svd() functions in Matlab

Assume we have a matrix X = randn(5,3). I am doing two things: 1) [S D1 V1] = svd(X); 2) ...
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### How to find eigenvalues and eigenvectors of the cokurtosis matrix?

Kurtosis is the fourth statistical moment of a random variable's distribution. Unlike the variance-covariance matrix $\Sigma$, which had a shape of $p\times p$, the kurtosis-cokurtosis matrix is ...
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### What's the importance of parallel eigenvectors?

I'm studying eigenvectors. I read that if a matrix is symmetric and if the eigenvalues are real numbers, the eigenvectors will be perpendicular. However, I have no idea what it means (if anything) ...
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### Are haar bases eigenfunctions for any kernel?

Are haar wavelet bases eigenfunctions for any kernel? If so, what Kernel is it, and how would we find the eigenvalues?
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### Angle between PCA vector spaces?

I have two datasets of the same shape, one for condition A, the other for condition B. I would like to test if the major axes of variance of condition A are different than those of B. Here is my idea. ...
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### Understanding the output from the Johansen Cointegration test in R

I have a VECM model that Im using to determine the revenues for a firm, based on factors like Interest rates, S&P 500 and company specific variables, as follows: Stage 1: $$z_t= a+ bX_t+e_t$$ ...
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### eigenstructure matching optimization

Is there any optimization loss functions that can approximately match the eigenstructure of the original samples and the transformed samples? For example, given a collection of samples $\mathbf{X}$ ...
Let $X$ be a random vector of dimension $p$ and $\{ X_1, \dots, X_n \}$ the $n$ observations of such vector. Let $\mathbb{X}$ be the matrix with rows $X_k$ and $\mathbb{X}_c$ is the matrix with rows \$...