# Questions tagged [eigenvalues]

For questions involving calculation or interpretation of eigenvalues or eigenvectors.

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### Does invariance of PCA under orthogonal transformation hold for data that is not centered?

I read the proof in the top answer to this question, but that page assumes that $\overline{A} = 0$. If the data instead has some nonzero mean $\mu$, I'm not sure if the same logic applies: ...
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### What is the variable v in this case? Is it standard deviation or variance?

The code below is from an example on the scikit-learn website for plotting the Gaussians of a GMM as ellipses. However, I don't know whether the final definition of the variable ...
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### Relation between generalization bounds of Kernel Ridge Regression and largest eigenvalue of the kernel Gram matrix

Consider a positive-definite, symmetric function $k(x_1, x_2)$ which is used, given the dataset $\{(x_i, y_i)\}_{i=1}^m$, to construct the Gram matrix $K = [k(x_i, x_j)]_{i,j \in 1, ..., m}$. What is ...
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### Are "weights", "loading scores", and "singular values" all synonyms?

I'm currently learning to use "eigenfaces" for facial image classification. Unfortunately, I've encountered some confusion with the following lines of code: ...
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### Show that the autocovariance matrix is positive definite

I've been working through the textbook Time Series Analysis and its Applications (R. H. Shumway & D. S. Stoffer 2ed). The topic I'm looking at is forecasting using ARMA models. The below assumes ...
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### Different eigenvalues output running EFA in SPSS VS R

I am new to R. I ran exploratory factor analysis (EFA) in SPSS and in R. The SPSS output suggests 3 factors of eigenvalue>1 (8.78;1.5;1.05), while R indicates there is just 1 factor with eigenvalue&...
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### The discrepancy of results of PCA via Eigendecomposition vs via SVD in Python with scipy.linalg [duplicate]

I recently learned about different methods of PCA. I decided to manually implement PCA in Python with Eigendecomposition of cov(X) and the Singular Value ...
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### Is a bivariate copula relevant in this physics setting manifesting uniform univariate marginals--and, if so, how can it be constructed?

To quickly place our probabilistic (copula) question in its subject matter setting, we note that a fundamental concept in quantum theory is that of entanglement QuantumEntanglement. The states of ...
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### All eigenvalues less than 1 in factor analysis

Is it possible to have all eigenvalues less than 1 during factor analysis? then what can be concluded here? since a factor is considered a factor when the eigenvalue of the factor is greater than 1. ...
1 vote
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### How do I get the stationary distribution of a Markov chain matrix from SVD?

I have a matrix that represents a Markov chain. ...
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### data projection on k - dime hyperplane (for eigenvector with largest eigenvalue) provides corrected representation

This is in reference to the text, outlier analysis by Charu C. Aggarwal: The author mentions, the projection of data points on the k - dimensional hyperplane corresponding to the largest eigenvalues (...
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### Eigenvalue decomposition

Why is the eigenvalue decomposition of the covariance matrix formula, $UΛU^T$ (assuming T means transpose) and not $UΛU^{-1}$? Where $U$ is eigenvectors and $Λ$ is a diagonal matrix consisting of ...
1 vote
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### Find first Principal Component (and loading) using a fast iterative algorithm without covariance matrix

I have a matrix $X$ and I would like to find its first principal component and the corresponding loadings. I would like to do this without computing the covariance matrix of $X$. How can I do so? This ...
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### Inequality on norm in terms of eigenvalue

I came across this inequality but not sure if it's true: $|\lambda_{\text{min}}|^2\|\hat{\beta}-\beta\|^2_2\leq (\hat{\beta}-\beta)'\hat{\Sigma}(\hat{\beta}-\beta)$ where $\lambda_{\text{min}}$ is the ...
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### Eigenvalues of time lagged covariance matrix - should be always real and positive?

Consider two random variables $X_t$ and $X_{t+\tau}$; where both come from random variable $X$ but are lagged by $\tau$. Assume that both are mean-free. If we want calculate covariance matrix of them, ...
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### number and size of eigenvectors in PCA

As I understand, the size of eigenvector produced in PCA should be min{n,N}, where N=number of samples and n=dimension of each sample (Right?). However, I have seen in couple of cases that this size ...
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### Making sure that the design matrix is positive (semi-) definite

In Bayesian linear regression, how do I make sure that the design matrix produced by a neural network $\Phi$ is positive definite? Computing the covariance matrix on the weight requires inverting --- ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
1 vote
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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 ...
1 vote
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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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### 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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