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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How to derive the three matrices of SVD from eigenvalue decomposition in Kernel PCA?

Kernel PCA is usually done via eigenvalue decomposition of the Kernel Matrix $\mathbf{K}$ and standard PCA via SVD of the input $\mathbf{X}$. In standard PCA as far as I know we can derive $\mathbf{S}...
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Confusion about principal component and major axis of the ellipse corresponding to the covariance matrix

Based on my understanding, in PCA, we try to find a linear combination of axes such that the variance in that direction is maximized. If variables have the covariance matrix $\Sigma$, then, the first ...
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The convergence of random variables to standard normal distribution

Let $V_s$ be $n\times s$ real matrix and consisting i.i.d $\mathcal{N}(0,1)$ random variables [*]. Suppose that $O_s^1$ is the orthogonal matrix, its first column being the normalization of the first ...
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Finding the vectors/modes "maximally responsible" for the fluctutation of a quantity?

I have a question which I fear might be simple but i am completely unable to figure out. Lets say we have a time-series of a vector of coordinates which define a molecule ( column vector of x,y,z) ...
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The relationship between eigenvalues of a covariance matrix and the variances of the same data matrix after using eigenvectors as bases

Suppose we have a data matrix $\mathbf{X}\in \mathbb{R}^{M\times N}$ with $M$ features, $N$ samples and zero means ($M \lt N$). The covariance matrix of $\mathbf{X}$ is $\mathbf{C_x}=\frac{1}{N}\...
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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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Generate a random covariance matrix with specified eigenspectra and diagonal elements and first off-diagonal?

I want to generate a random covariance matrix ($c \in \mathcal{R}^{n \times n} $) whose eigenspectra, i.e., $n$ eigenvalues $e_0 \in \mathcal{R}^{n\times 1}$ and diagonal elements $c_{ii} \,\, i=1 \,\,...
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Eigenvalues of Johansen Trace Test

I'm currently taking a course in time series and have been struggling with understanding the Johansen trace test. Specifically, the calculation of the eigenvalues for the Likelihood ratio statistic. ...
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EFA: Eigenvalues or Loadings after extraction (SPSS)

I've been doing a EFA with ML extraction and Promax rotation, whereby three factors were extracted. For reporting the results, I was wondering whether to use the 'Initial Eigenvalues' or the '...
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Is it possible to derive the expected sampled eigenvalues given a population correlation matrix?

My question is : Is it possible to derive the expected sampled eigenvalues given a population correlation matrix and a sample size $n$. I am pondering about how behaves eigenvalues of correlation ...
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Popular symmetric positive definite matrices with explicitly known eigenvalues (at least largest eigenvalue)

I am looking for popular symmetric positive definite matrices with explicitly known eigenvalues (at least largest eigenvalue) arising as autocovariance matrices in time series (for example). In fact, ...
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Why is sum of squares equal to eigenvalue in PCA?

We fit a line or a hyperplane to a set of points. We project the points onto the hyperplane. The sum of squared distances of the projected points to origin is equal to the eigenvalue. Why is that?
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Positive semi definite matrix with negative eigenvalues?

From what I know, for any square real matrix A, a matrix generated with the following should be a positive semidefinite (PSD) matrix: ...
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Why in PCA the sum of the distances of from the principal component is the explained variance?

I am trying to understand why, when I calculate the principal components, summing the distances of the original points to the best fit line (eigenvector) equals the variance this PC explains?
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Eigenvector centrality comparison

Let's say we calculate the eigenvector centrality of the same set of nodes in different years (we have a network each year). Note that the eigenvector centrality is normalized in such a way that the ...
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Finding the eigenvalues of covariance matrix [duplicate]

Came across this as an interview question that I saw online: given a covariance matrix with diagonals being all 1 and the off-diagonals being c, what are its eigenvalues? Going by the definition $Av = ...
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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. ...
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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 ...
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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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How to calculate eigen-decomposition for a positive kernel in R^2 using R

Suppose $K$ is a continuous positive semi-definite kernel on $\mathcal{T} = [0,1]\times[0,1] \subseteq \mathbb{R}^2$. By Mercer Theorem, there is a eigen-decomposition of $K$ such that $$ K(x,y) = \...
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Negative Eigenvalues in EFA

Assuming a set of data meet all assumptions for EFA and we are doing a factor analysis (with the SMC used to define the shared variance), how do negative eigenvalues appear and what does that say ...
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Why is the first canonical direction equal to the left singular vector i.e. why is $w_1 = a = u_1$ in CCA (Canonical Correlation Analysis)?

I want to understand why the canonical direction $a$ is equal to the left singular value of $M = \Sigma^{-1/2}_X \Sigma_{X, Y} \Sigma^{-1/2}_Y$ and not $a = \sigma_1 u_1$. My calculation tell me that ...
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Relation between PCA eigen values and data visualization

We have matrix data $X$ which is $n\times d$. We use the covariance matrix/ design matrix/ gram matrix $X^T X$ to perform least-squares/ PCA. I compute the eigen basis representation of said matrix $$...
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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 ...
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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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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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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 ...
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