Power Method We now describe the power method for computing the dominant eigenpair. Its extension to the inverse power method is practical for finding any eigenvalue provided that a good initial approximation is known. Some schemes for finding eigenvalues use other methods that converge fast, but have limited precision. The inverse power method is then invoked to refine the numerical values and...

We propose a principled account on multiclass spectral clustering. Given a discrete clustering formulation, we first solve a relaxed continuous optimization problem by eigendecomposition. We clarify the role of eigenvectors as a generator of all optimal solutions through orthonormal transforms. We then solve an optimal discretization problem, which seeks a discrete solution closest to the conti...

For a large class of linear neutral type systems the problem of eigenvalues and eigenvectors assignment is investigated, i.e. finding the system which has the given spectrum and almost all, in some sense, eigenvectors.

Eigenvector continuation is a computational method that finds the extremal eigenvalues and eigenvectors of Hamiltonian matrix with one or more control parameters. It does this by projection onto subspace corresponding to selected training values The has proven be very efficient accurate for interpolating extrapolating eigenvectors. However, almost nothing known about how converges, its rapid co...

Eigenvectors of large matrices (and graphs) play an essential role in combinatorics and theoretical computer science. The goal of this survey is to provide an up-to-date account on properties of eigenvectors when the matrix (or graph) is random.

In many applications, one has side information, e.g., labels that are provided in a semisupervised manner, about a specific target region of a large data set, and one wants to perform machine learning and data analysis tasks “nearby” that prespecified target region. For example, one might be interested in the clustering structure of a data graph near a prespecified “seed set” of nodes, or one m...

We discuss an algebraic method for constructing eigenvectors of the transfer matrix of the eight vertex model at the discrete coupling parameters. We consider the algebraic Bethe ansatz of the elliptic quantum group Eτ,η(sl2) for the case where the parameter η satisfies 2Nη = m1 + m2τ for arbitrary integers N , m1 andm2. Whenm1 orm2 is odd, the eigenvectors thus obtained have not been discussed...

In this paper the HDWTSVD algorithm to encode monochromatic images is proposed. The algorithm combines DWT and SVD techniques. The input image is divided into tiles of 64x64 pixels. A criterion based on the average standard deviation of 8x8 subblocks is used to choose DWT or SVD. If the tile exhibits a high average standard deviation, it is compressed by using SVD otherwise by DWT. Eigenvalues ...

Motivated by the conjectures formulated in 2003 [28], we study interlacing properties of eigenvalues A⊗B+B⊗A for pairs n-by-n matrices A,B. We prove that every pair symmetric (and skew-symmetric matrices) with one them at most rank two, odd spectrum (those determined eigenvectors) interlaces its even eigenvectors). Using this result, also show when n≤3, The results specify structure eigenvector...

Spectral clustering is a well-regarded subspace algorithm that exhibits outstanding performance in hyperspectral image classification through eigenvalue decomposition of the Laplacian matrix. However, its accuracy severely limited by selected eigenvectors, and commonly used eigenvectors not only fail to guarantee inclusion detailed discriminative information, but also have high computational co...