Principal Component Analysis (PCA) is a standard tool for dimensional reduction of a set of n observations (samples), each with p variables. In this paper, using a matrix perturbation approach, we study the non-asymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite sample of size n, to those of the limiting population PCA as n → ∞. As in machine learning, we pr...

Eigenvalue spectrum of adjacency matrices of many complex networks reveals that a large real eigenvalue separate from the bulk of the population of eigenvalues. A related theorem in this observation is that of Perron-Frobenius, which states that ifˆA is a positive matrix, then there exists a unique eigenvalue ofˆA, which has the greatest absolute value, and its associated eigenvector may be tak...

Principal component analysis (PCA) is a standard tool for dimensional reduction of a set of n observations (samples), each with p variables. In this paper, using a matrix perturbation approach, we study the nonasymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite sample of size n, and those of the limiting population PCA as n→∞. As in machine learning, we pres...

We analyze the largest eigenvalue and eigenvector for the adjacency matrices of sparse random graph. Let λ1 be the largest eigenvalue of an n-vertex graph, and v1 be its corresponding normalized eigenvector. For graphs of average degree d log n, where d is a large enough constant, we show λ1 = d log n + 1 ± o(1) and 〈1, v1〉 = √ n ( 1−Θ ( 1 logn )) . It shows a limitation of the existing method ...

This paper generalizes the well-known identity which relates the last components of the eigenvectors of a symmetric matrix A to the eigenvalues of A and of the matrix An−1, obtained by deleting the last row and column of A. The generalizations relate to matrices and to Sturm–Liouville equations.

There are methods to compute error bounds for a multiple eigenvalue together with an inclusion of a basis of the corresponding invariant subspace. Those methods have no restriction with respect to the structure of Jordan blocks, but they do not provide an inclusion of a single eigenvector. In this note we first show under general assumptions that a narrow inclusion of a single eigenvector is no...

In this note we prove a conjecture from [DFJMN] on the asymptotics of the composition of n quantum vertex operators for the quantum affine algebra Uq(ŝl2), as n goes to ∞. For this purpose we define and study the leading eigenvalue and eigenvector of the product of two components of the quantum vertex operator. This eigenvector and the corresponding eigenvalue were recently computed by M.Jimbo....

The spectral properties of the adjacency matrix, in particular its largest eigenvalue and the associated principal eigenvector, dominate many structural and dynamical properties of complex networks. Here we focus on the localization properties of the principal eigenvector in real networks. We show that in most cases it is either localized on the star defined by the node with largest degree (hub...

This paper establishes converses to the well-known result: for any vector ũ such that the sine of the angle sin θ(u, ũ) = O( ), we have ρ(ũ) def = ũ∗Aũ ũ∗ũ = λ+O( ), where λ is an eigenvalue and u is the corresponding eigenvector of a Hermitian matrix A, and “∗” denotes complex conjugate transpose. It shows that if ρ(ũ) is close to A’s largest eigenvalue, then ũ is close to the corresponding ei...