# Questions tagged [eigenvalues]

For questions involving calculation or interpretation of eigenvalues or eigenvectors.

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### How to quantify the similarity of EOF loading in multiple matrices?

I have five 3-D matrices (time, latitude, and longitude) representing the same variable but from different sources, denoted as A, B, C, D, and E. I calculated the first five EOF loadings for each of ...
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### Matrix GLM estimation Example Explanation

**Below is my question: It is an example from the book " A Primer on linear modes" by Monahan** I have 3 questions Question 1: is how did he calculated the matrix λ Question 2: Why $λ$ ...
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I wanted to confirm my intution about the meaning of "loadings" I have made out of eigenvalue/eigenvector decomposition, but I still fail to do that. Note that I made 3 pairs of highly correlated ...
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### Scatter Plot - Basics [closed]

I am stuck in understanding a basic scatter plot. I am working in two dimensions i.e. there are two variables X & Y. So, the question is that in the scatter plot, what do the two axes mean? ...
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### How can I adjust for negative eigenvalues?

I wish to run a path analysis from a pooled correlation matrix that I have imputed using the maximum-likelihood procedure. There was considerable missing data. The resulting correlation matrix is: <...
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### Using NbClust on datasets that produce some negative eigenvalues. When to exclude data, when to force to positive, when to exclude test index?

Background on why I am using clustering: I am analyzing data from a multistep biological experiment, where each step is done in batches of varying sizes. I want to account for any biases that might ...
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### Why are the eigenvalues of $X'X$ equal to that of $XX'$ when $X$ is a design matrix? [duplicate]

The title says it all. If $X$ is a design matrix (columns containing variables, rows containing observations), I have observed that eigs($X'X$)=eigs($XX'$). I actually found this by accident when I ...
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### How to apply Principal components (Eigenvectors) in PCA?

I struggle to understand how to get further in my PCA analysis after computing the eigenvectors of the covariance matrix. I choose the number of principal components which contain 95% of the ...
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### Is the first principal component is the one with the largest eigenvalue and how to convert it to explained variance?

In PCA, after we calculate the eigenvalues of each variable, we need to get the explained variance, I read an article which suggests: ...
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### Procedure to quickly find the near zero eigenvalues (and corresponding eigenvectors) of a positive semidefinite square matrix?

I have an ill-conditioned positive semidefinite $n \times n$ square matrix (Hessian), with unknown rank $r$, that I need to compute the inverse for. I'd prefer not to compute the SVD for performance ...
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### Gradient descent derivation in Eigenspace [duplicate]

I am trying to decode article on https://distill.pub/2017/momentum/ I was able to follow everything until the part with a change of basis x$^k=Q^T(w^k−w^⋆)$ to eigenspace... I conceptually understand ...
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### Eigenvalue decomposition of a covariance matrix using a fast Cholesky decomposition

Let $\mathbf{C}$ be a $n \times n$ covariance matrix and assume that the LDL' Cholesky decomposition can be obtained efficiently. Can we take advantage of this to obtain a fast eigenvalue ...
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### Does $\text{cov}(a_1' X, a_2' X) = 0$ imply $a_1 \cdot a_2 = 0$?

Let $X$ be a $p$-dimensional random vector with $p$ principal components $y_1, y_2, \dots, y_p$. By definition, a restriction put on the second principal component $y_2 = a_2'X$ is  \text{cov}(y_1, ...
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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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### Calculate first principal component direction and scores

Given that x1 = (9, 9, −18)^T and x2 = (18, 9, 9)^T with eigendecomposition of its sample covariance matrix Σ = cov(X) How do I calculate the first two principal component direction and the ...
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### Failure to replicate calculation of PCA residuals in linear regression with heteroscedasticity

In their preprint, Rocha et al. suggest a new type of residual for linear regression models with heteroscedasticity. They call their new residual PCA residuals. I have tried to replicate some of their ...
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### Is it right to use PCA in this scenario?

Physicist here. I have a dataset. The data is the emission from a molecule that has two dipoles. Molecules can only emit along these dipoles. As I rotate the molecule, I will selectively excite the ...
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### Is eigenspace based classification possible

Imagine I would like to classify an image (e.g. into healthy and sick) and have a lot of labeled data. Could I classify any image by comparing it to the eigenspaces of the two sets? It sounds simple, ...
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### Factor analysis

I have a couple of questions on factor analysis using Stata. How to decide whether to use pf (principal factors, default), pcf (principal-components factor), ipf (iterated principal factor) or mle (...
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### When the elements of first basis are always positive for PCA?

I am computing the PCA projection matrix of some data. I notice that the elements of first basis vector (corresponding to the highest eigenvalue) are always positive. My data is real and contain both ...
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### Is it better to interpret PCA components using the eigenvectors or the rescaled loadings?

I have a dataset to which I am applying PCA, and looking to each PCA component. Initially I was using the eigenvectors as a way to understand what each component "means". When using the eigenvectors ...
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### eigen function in R

I want to ask about the eigen function of R. I am currently doing a project of NBA team analysis. I am trying to figure out correlation effect of two players lineup ...
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### Basis vectors for categorical images

I have a sequence of categorical images. For a two category image, each image pixel can have one of two values. I would like to analyze these images using a technique like eigen images. The goal is to ...
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### How to interpret eigenvectors in PCA analysis?

I'm trying to apply the output from PCA analysis I've run on some yield curve history and am getting a bit confused. I have followed the steps below, From a history of the yield curve ($m \times n$ ...
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### Kaiser Criterion after Rotation

Within my PCA, I rotated the solution, so the output would make more sense given the complex structure using a Varimax rotation. However, when I am confirming the number of factors to extract based on ...
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### Drawing 95% ellipse over scatter plot

The context is regression analysis using Eviews, but first I wanted to create a few scatter plots and overlay error ellipses on them. Eviews doesn't support that kind of graph ornamentation so I am ...
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### Significance of eigenvector components in PCA

Long time reader, first time poster. Hopefully I won't screw this up... In the context of Principal Component Analysis, I have the sense that the components of an eigenvector are a measure of the ...
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### adding a small constant to the diagonals of a matrix to stabilize

I have a large correlation matrix (110x110) with some small eigenvalues (about 20 < 0.1). It has been suggested that adding a constant (about 0.1) to the diagonals will help to stabilize the matrix....
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### Graphical understanding of PCA

I learned about PCA and how to find the principal components via eigenvectors/values. Now for the following problems my professor says that "Feature 2 is constant and can hence be ignored, so you can ...
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### importance of correlation between data for a PROC CLUSTER

i'm working on a clustering analysis on SAS. I need to improve an actual code : ...
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### Determining the Direction of Eigenvectors in PCA [duplicate]

I'm using R to get the principal components for several datasets. An example result, using prcomp yields: ...
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