# Questions tagged [eigenvalues]

For questions involving calculation or interpretation of eigenvalues or eigenvectors.

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### Error term in SGD with momentum

I am reading the article "How Momentum really works" (https://distill.pub/2017/momentum/), and i am confused in one point: I am trying to derive the convergence rate for momentum from the ...
• 121
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### Why do I need Loadings when I can reconstruct data with eigenvectors in PCA?

I have read the responses to this question here, here and here, but I am still confused on the application of loadings and eigenvectors. Principally this statement (from the first link), It is ...
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### Linear algebra properties of a confusion matrix (eigenvalues, eigenvectors, and determinants)

This answer to a question on Math Stack Exchange got me thinking about a confusion matrix as more than just a rectangular array of numbers. We don’t talk about a confusion matrix as a linear ...
• 62.6k
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### Compare Eigenvector Centrality of two networks

We have two networks (G1, G2), one with 4 times more nodes than the other, and we want to compare the eigenvector centrality of their three most central nodes (e.g. top_1 node of G1 vs top_1 node of ...
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### Are squared loadings and squared distances the same in Principal Component Analysis?

I read online that: Eigenvalue: Represents the variance explained by the principal component. It equals both the sum of squared loadings for that component and the sum of squared projections of data ...
85 views

### Expected value of largest eigen value of sample correlation matrix

Suppose $X$ follows some multivariate distribution with zero mean and Identity covariance matrix. Suppose $X$ is N dimensional. Suppose $R$ is the sample correlation matrix, calculated based on n ...
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### Distribution of Maximum Eigen Value

Suppose I have X, k*n, where $M=X'X$. Suppose $n>>k$, and $rank(M) =k-1$. Suppose $\lambda_1, \cdots, \lambda_{k-1}$ are the eigen values of M. Under the assumption that the columns of X are ...
• 265
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### Interpreting eigenvalues of non-normalized covariance matrix of physical system

Cross-posted from physics stackexchange Summary: Eigenvalues of a "non-normalized" covariance matrix of time-series measurements from a linear system have units of Action (energy * time). ...
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### The eigen values of Johansen's cointegration procedure

Assume a K dimension VECM model for cointegration analysis $$\Delta y_t=\Pi y_{t-1}+\Gamma_1\Delta y_{t-1}+...+\Gamma_{p-1}\Delta y_{t-p+1}+u_t$$ The Johansen approach for maximum eigenvalue test or ...
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### Using sample covariance to estimate eigenvalues of population covariance?

Suppose $X$ is an $m\times n$ data matrix with $m\approx \infty$ and $X_k$ consists of random $k$ rows of $X$. How do I estimate the spectrum of $X$ from $X_k$? I can only reliably estimate $K$ ...
• 6,219
1 vote
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### How to balance PCA and LDA in subspace learning?

PCA is a generative model, by which input images or data can be reconstructed. LDA (Linear Discriminant Analysis) is a discriminative model, which extracts better features for classification. How to ...
61 views

### Solving an exercise about admissible coefficient values for a MA(1) process

I'm studying "Principles of system identification : Theory and Pratice" by Arun K. Tangirala and well... I've just entered the part about moving averages and I'm confused. I don't understand ...
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1 vote
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### Why do eigenvalues of $\mathbf\Phi^T\mathbf\Phi$ increase with the size of data set?

The question comes from a paragraph in page 171 of "Pattern Recognition and Machine Learning" by Christopher M. Bishop: Here $\mathbf\Phi$ is the design matrix for a data set of $N$ samples ...
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### Why is $\mathbf\Phi^{\top}\mathbf\Phi$ a positive definite matrix?

I had this question when reading section 3.5.3 on page 170 of "Pattern Recognition and Machine Learning" written by Christopher M. Bishop: Here $\mathbf\Phi$ represents the design matrix ...
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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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1 vote