# Questions tagged [eigenvalues]

For questions involving calculation or interpretation of eigenvalues or eigenvectors.

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### Making sure that the design matrix is positive (-semi) definite

In bayesian linear regression, how to make sure that the design matrix produced by a neural network $\Phi$ is positive definite? Because to computing the covariance matrix on the weight requires ...
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### How does eigendecomposition form principal components? [duplicate]

I have a few questions regarding how specifically principal components are formed: What is the relevance of the magnitudes of the covariance when it comes to eigendecomposition? How does ...
187 views

### Get accurate eigenvectors, when eigenvalues are minuscule

I have a symmetric matrix A. I'm not able to compute all the eigenvectors accurately, and I believe it is due to the last few eigenvalues for ...
50 views

### What are the metrics to assess the quality of a multiple correspondence analysis (MCA) model?

We are trying to implement a multiple correspondence analysis (MCA) model. I was looking for metrics to assess the quality of an MCA to evaluate our model. Sadly, I didn’t find much literature about ...
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### How to interpret a PCA Biplot? [duplicate]

I constructed a PCA plot from a very high-dimensional dataset that contains features relating to fraud. After creating the PCA plot, I created a biplot with the features to see how they interact. The ...
47 views

### Why absolute value of eigenvalues are used in PCA or LDA?

In PCA and LDA techniques, eigenvectors with the $k$ largest eigenvalues give principal components. However, when selecting these eigenvalues, are they to be sorted by the absolute value (regardless ...
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### What does it mean for Katz Centralities to "diverge"

In Mark Newman's Networks book, 2010 edition, page 173, he explains some mathematical details behind the Katz Centrality measure: In matrix terms, Eq. (7.8) can be written x = αAx + β1, (7.9) where 1 ...
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### why do we need orthogonal basis for PCA? [duplicate]

In PCA, why do we map to the orthogonal basis? What is the important point that they should be orthogonal with each other?
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### Are eigenvectors of PCA guaranteed to be orthonormal?

Are eigenvectors (principal components) of PCA orthonormal or only orthogonal ? Or only some of them are orthonormal or they are orthonormal if data were normalized before doing PCA ?
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### Does the Cholesky decomposition of a covariance matrix lead to a lower triangular matrix with positive diagonals?

We know that an $N\times N$ covariance matrix $\Sigma$ is symmetric positive definite, and can be factorized using Cholesky decomposition as follows \begin{equation} \Sigma=LL' \end{equation} where $L$...
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### Normalizing a custom Weight Shifted or Spiked Gaussian distribution

I have a custom weight shifted bivariate gaussian distribution that I wish to normalize. W is the weighted symmetric matrix that shifts the entire distribution and the λ below is the diagonal matrix ...
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### Very low values of explained variance for the first Axes from PCoA

I am comparing different sites based on their floristic composition in R. Therefore, I have created a huge community datamatrix (presence/absence data) from 53 sites including over 1000 species. To ...
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### In the rotational cost algorithm, are eigenvectors sorted according to eigenvalues, or according to absolute value of eigenvalues?

I am trying compute Rotational Cost, as defined in the 2004 paper "Self-Tuning Spectral Clustering" by Zelnik-Manor and P. Perona (http://www.vision.caltech.edu/lihi/Publications/...
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### Disambiguation of spectral clustering

I've read about spectral clustering, and feel like there are two related but different ways people use the words "spectral clustering". Way 1 Spectral clustering as a pre-processing step ...
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### How to check if my matrices are 'random' using Marchenko-Pastur?

For each $n=1,2,\dots,100$, I have an $r \times s$ integer-valued matrix $X(n)$ of increasing dimension as $n$ increases. That is, the number of rows $r$ and columns $s$ increases as $n$ increases. ...
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### Is it possible that Fisher information matrix be indefinite? [duplicate]

I`m using the Newton-Raphson method for obtaining MLE for parameters for maximizing my objective function. At each iteration, I want to check that is the Hessian matrix negative definite or not and I ...
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### Why must the non-zero eigenvalues in the Johansen test be between 0 and 1?

Why are the non-zero eigenvalues in the matrix $\Pi$ in the Johansen test between 0 and 1? Why can't they be greater than 1 or less than zero? My lecturer just dropped in that's its because "the ...
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### PCA face recognition (problem) faces are not recognized

I need advice regarding PCA face recognition. I have a dataset of 86 faces, which I stretch into a column vector. (so my matrix is 3000 by 86, where columns are faces). I calculate the average face ...
31 views

### LDA (linear discriminant analysis) for images: strange eigenvalues

I have the following dataset: I represent each image as a $(67 \times 67, 1)$ vector and add it to the dataframe. df.head() My goal is to determine the ...
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### Is the QR Algorithm guaranteed to compute eigenvectors?

I'm writing some C++ matrix library for hobby. For computing eigenvalues and eigenvectors, I referred the following "Francis double step QR algorithm": In particular, page 82 of https://...
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### Is there any relationship between the mode value of a data set and the main component in PCA for this dataset?

I'm trying to intuitively understand if there is any direct relationship between the mode value of a dataset, which represents the value that repeats the most, and the main PCA component for this same ...
80 views

### MANOVA: How to identify the most relevant dependent variables

I am facing a problem with a big number of dependent variables and relatively small sample size. Not all dependent variables might be relevant, though. As multivariate analysis of variance (MANOVA) ...
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### Alternative of the Nyström method

Say you want to obtain the eigenvalues/vectors of the integral operator associated to a kernel $K$. I know there is the Nyström method to obtain an approximation of these eigenvalues/vectors. What are ...
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### Mercer's theorem and eigenfunctions

Consider a kernel $K$ which satisfies the conditions of Mercer's theorem. We know that $K(x,y) = \sum\limits_{i=1}^{+\infty}\lambda_ie_i(x)e_i(y)$ where $\lambda_i$ and $e_i$ are the eigenvalues/...
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### How to translate eigenvectors and eigenvalues to the number of clusters in spectral clustering?

I have generated this output, where L is the Laplacian Matrix, D is the degree and A is the adjacency matrix: I can see the eigenvalues and eigenvectors are returned. I am unsure how to interpret ...
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### How to calculate the eigenvector?

I've been struggling to solve this math problem for two days... So I calculated the mean of all samples (0,0). Put it into the equation and got V as \begin{array} {rrr} 4 & 2 \\ 2 & 2 \end{...
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### Why is sum of squared PCA column loadings equal to corresponding Eigenvalue?

I'm wondering why or how is the sum of squared PCA loadings in the column equal to Eigenvalue. I understood that the sum of squared PCA loadings in the row is equal to 1 or 100% because original ...
289 views

### How to use principal components as inputs in hierarchical clustering analysis in R

For my statistical analysis I want to follow the steps of a paper I read. I have a dataset in which each row corresponds to a dive carried out by a whale ('id' in table below) and the columns to the ...
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### Why is there a discrepancy between the eigenvalues of the covariance matrix (PCA) and the eigenvalues of the kernel matrix (kernel PCA)?

I've done PCA on my data matrix $\mathbf{X}$ which gives me i.a. the eigenvalues $\lambda$ and eigenvectors $v$ of the data covariance matrix $C=\mathbf{X}^T \mathbf{X}$. I'm now extending my ...
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### Can you sum principal components?

I am currently reviewing principal components from my recent output. For each principal component I know you get an eigenvalue which represents how much of the variance is explained. If I was to take ...
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### Why 'eigen()' and 'fa.parallel()' give different eigenvalues? [closed]

I ran an EFA on 10 items in Rstudio. I did parallel analysis (fa.parallel) using psych package and also eigenvalues using these ...
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### Doubt in the range of variance of First Principal Component

This is my first problem on this forum. Here is the problem: The covariance matrix of a four dimensional random vector $\boldsymbol X$ is of the form \begin{bmatrix}1&\rho&\rho&\rho\\ \...
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### How to recover true eigenvectors after creating data matrix from synthetic PCA results?

I try to create a raw data matrix given a matrix of eigenvectors and some synthetic principle components and want to discover the true eigenvectors. Assume I have the following random data matrix: <...
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### What is the precise relation between the eigenvalues of a covariance *function* and the eigenvalues of a covariance *matrix*?

Assume we have a temporal Gaussian Process $\mathcal{GP}(t;\ m,k)$ (GP) with mean $m$ and covariance function (aka. kernel) $k$ on some compact time interval $[0,T]$. Then, the eigenvalues $\lambda$ ...
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### Multivariate normal distribution - diagonalizable matrix

Let $X_{1},..,X_{n}$ be independent random vectors drawn from the $N_{p}(\mu ,\Sigma )$ distribution and let $\bar{X}$ be their mean. Find a $p\times np$ matrix $A$ and vector $v\in \mathbb{R}^{np}$ ...
### OLS estimator is consistent if the smallest eigenvalue of $X^TX$ goes to infinity as $n\to\infty$
I want to show that if $\lambda_{min}(X^T X)$ (i.e., the smallest eigenvalue of $X^TX$) goes to infinity as $n\to\infty$, then $\hat{\beta}$ is a consistent estimator of $\beta$. My approach is the ...
I have recently read a paper where the authors applied PCA to determine the weights of the variables used to calculate a composite index. In the methodology, they mentioned that for a set of $N$ ...