If a set of independent, identically distributed random vectors has heavy tails, so that the covariance matrix does not exist, there is no reason to expect that the sample covariance matrix conveys useful information. On the contrary, this paper shows that the eigenvalues and eigenvectors of the sample covariance matrix contain detailed information about the probability tails of the data. The e...

A modification is given of the GMRES iterative method for nonsymmetric systems of linear equations. The new method deflates eigenvalues using Wu and Simon’s thick restarting approach. It has the efficiency of implicit restarting, but is simpler and does not have the same numerical concerns. The deflation of small eigenvalues can greatly improve the convergence of restarted GMRES. Also, it is de...

We study the limiting eigenvalue distribution of n×n banded Toeplitz matrices as n → ∞. From classical results of Schmidt-Spitzer and Hirschman it is known that the eigenvalues accumulate on a special curve in the complex plane and the normalized eigenvalue counting measure converges weakly to a measure on this curve as n → ∞. In this paper, we characterize the limiting measure in terms of an e...

We study the $\mathfrak{gl}_{m|n}$ XXX spin chains defined on tensor products of highest $\mathfrak{gl}_{m|n}$-modules. show that on-shell Bethe vectors are eigenvectors higher transfer matrices and compute corresponding eigenvalues, confirming Conjecture 5.15 arXiv:2007.15573 extending main results arXiv:0605015 to supersymmetric case. then take classical limits obtain for Gaudin models.

If you were to ask a controls engineer why one should care about eigenvalues, the answer would be obvious and immediate. Eigenvalues are absolutely critical anyone designing controllers or wanting understand how system behaves. can literally tell us whether not an aircraft will crash, if infectious disease turn into global pandemic. Now, same question college student early on in their education...

1 Useful Linear Algebra Let v = (v1, v2, . . . , vn) be a non-zero n-dimensional row vector and P an n× n matrix. • We say v is an eigenvector of P with corresponding eigenvalue λ iff vP = λv. • The L1-norm of v (denoted ‖v‖1) is ∑n i=1 vi. • The L2-norm of v (denoted ‖v‖2) is √∑n i=1 v 2 i . • The inner product of two vectors v and w (denoted v ·w) is ∑n i=1 viwi. • We say vectors v,v, . . .v ...

Noise variance estimation is required in many image denoising, compression, and segmentation applications. In this work, we propose a fast noise variance estimation algorithm based on principal component analysis of image blocks. First, we rearrange image blocks into vectors and compute the covariance matrix of these vectors. Then, we use Bartlett’s test in order to select the covariance matrix...

The Lanczos process is a well known technique for computing a few, say k, eigenvalues and associated eigenvectors of a large symmetric n×n matrix. However, loss of orthogonality of the computed Krylov subspace basis can reduce the accuracy of the computed approximate eigenvalues. In the implicitly restarted Lanczos method studied in the present paper, this problem is addressed by fixing the num...

The Gamow vector description of resonances is compared with the S-matrix and the Green function descriptions using the example of the square barrier potential. By imposing different boundary conditions on the time independent Schrödinger equation, we obtain either eigenvectors corresponding to real eigenvalues and the physical spectrum or eigenvectors corresponding to complex eigenvalues (Gamow...

We present a prediction and regularization strategy for alleviating the conventional problems of LDA and its variants. A procedure is proposed for predicting eigenvalues using few reliable eigenvalues from the range space. Entire eigenspectrum is divided using two control points, however, the effective low-dimensional discriminative vectors are extracted from the whole eigenspace. The estimated...