The generalized minimum residual method (GMRES) is well known for solving large nonsymmetric systems of linear equations. It generally uses restarting, which slows the convergence. However, some information can be retained at the time of the restart and used in the next cycle. We present algorithms that use implicit restarting in order to retain this information. Approximate eigenvectors determ...

The n-dimensional numerical range of a densely defined linear operator T on a complex Hilbert space H is the set of vectors in Cn of the form (〈Te1, e1〉, . . . , 〈Ten, en〉), where e1, . . . , en is an orthonormal system in H, consisting of vectors from the domain of T . We prove that the components of every corner point of the n-dimensional numerical range are eigenvalues of T .

This article gives a short tutorial on the MUSIC algorithm and the linear sampling method of [3], and explains how the latter is an extension of the former. 1 MUSIC MUSIC is an abbreviation for MUltiple SIgnal Classification [10]; we see below why the name is appropriate. 1.1 The basics of MUSIC MUSIC is essentially a method of characterizing the range of a self-adjoint operator. Suppose is a s...

In this paper, we represent protein structure by using graph. A protein structure database will become a graph database. Each graph is represented by a spectral vector. We use Jacobi rotation algorithm to calculate the eigenvalues of the normalized Laplacian representation of adjacency matrix of graph. To measure the similarity between two graphs, we calculate the Euclidean distance between two...

Recall this theorem from last time. Theorem 1 (Singular Value Decomposition and best rank-k-approximation) An m × n real matrix has t ≤ min {m,n} nonnegative real numbers σ1, σ2, . . . , σt (called singular values) and two sets of unit vectors U = {u1, u2, . . . , ut} which are in <m and V = v1, v2, . . . , vt ∈ <n (all vectors are column vectors) where U, V are orthogonormal sets and ui M = σi...

A real n-dimensional homogeneous polynomial f (x) of degree m and a real constant c define an algebraic hypersurface S whose points satisfy f (x) = c. The polynomial f can be represented by Axm where A is a real mth order n-dimensional supersymmetric tensor. In this paper, we define rank, base index and eigenvalues for the polynomial f , the hypersurface S and the tensor A. The rank is a nonneg...

the analysis of cross-correlations is extensively applied for understanding of interconnections in stock markets. variety of methods are used in order to search stock cross-correlations including the random matrix theory (rmt), the principal component analysis (pca) and the hierachical ‎structures.‎ in ‎this work‎, we analyze cross-crrelations between price fluctuations of 20 ‎company ‎stocks‎‎...

The missing label for basis vectors of [Formula: see text] representations corresponding to the reduction can be provided by eigenvalues scalars in enveloping algebra text]. There are only two such independent elements degrees three and four. It is shown how one degree four diagonalized using analytical Bethe ansatz.

It can be reduced to a product of N one dimensional integrals by diagonalizing the matrix K ≡ Ki,j . Since we need only consider symmetric matrices (Ki,j = Kj,i), the eigenvalues are real, and the eigenvectors can be made orthonormal. Let us denote the eigenvectors and eigenvalues of K by q̂ and Kq respectively, i.e. Kq̂ = Kqq̂. The vectors {q̂} form a new coordinate basis in the original N dimensi...

We introduce a new random matrix model called distance covariance in this paper, whose normalized trace is equivalent to the covariance. first derive deterministic limit for eigenvalue distribution of when dimensions vectors and sample size tend infinity simultaneously. This valid are independent or weakly dependent through finite-rank perturbation. It also universal details distributions vecto...