Showing papers by "Roger A. Horn published in 2006"
TL;DR: Canonical forms for congruence and ∗congruence of square complex matrices were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347-353] as discussed by the authors.
92 citations
TL;DR: In this paper, a regularization algorithm that uses only elementary row operations to construct such a decomposition is presented. But it is only applicable to real orthogonal transformations and a reduced form that can be achieved via a unitary *congruence or congruence.
34 citations
TL;DR: A complex analog of PageRank is obtained for the web hyperlink matrix G(c) with a complex parameter c with regularity, limits, expansions, and conditioning of y(c), and a complex extrapolation algorithm is proposed that may provide an efficient way to compute PageRank.
Abstract: The spectral and Jordan structures of the web hyperlink matrix G(c) = cG + (1 - c)evT have been analyzed when G is the basic (stochastic) Google matrix, c is a real parameter such that 0 < c < 1, v is a nonnegative probability vector, and e is the all-ones vector. Typical studies have relied heavily on special properties of nonnegative, positive, and stochastic matrices. There is a unique nonnegative vector y(c) such that y(c) TG(c) = y(c) T and y(c) T e = 1. This PageRank vector y(c)can be computed effectively by the power method. We consider a square complex matrix A and nonzero complex vectors x and v such that Ax = λx and v*x = 1. We use standard matrix analytic tools to determine the eigenvalues, the Jordan blocks, and a distinguished left λ-eigenvector of A(c) = cA + (1 - c)λxv* as a function of a complex variable c. If λ is a semisimple eigenvalue of A, there is a uniquely determined projection P such that lim c→1 y(c) = Pv for every v such that v*x = 1; this limit may fail to exist for some v if λ...
32 citations
01 Jan 2006
TL;DR: In this paper, the PageRank vector (the normalized dominant left eigenvector of G(c)) is computed in the Schur triangular form and the eigenvalues of A(c) are λ, cλ2,..., cλn.
Abstract: Let A be a given n-by-n complex matrix with eigenvalues λ, λ2, . . . , λn. Suppose there are nonzero vectors x, y ∈ Cn such that Ax = 3Dλx, y∗A = 3Dλy∗, and y∗x = 3D1. Let v ∈ Cn be such that = v∗x = 3D1, let c ∈ C, and assume that λ 6= cλj for each j = 3D2, . . . , n. Define A(c) := 3DcA + (1 − c)λxv∗. The eigenvalues of A(c) are λ, cλ2, . . . , cλn. Every left eigenvector of A(c) corresponding to λ is a scalar multiple of y − z(c), in which the vector z(c) is an explicit rational function of c. If a standard form such as the Jordan canonical form or the Schur triangular form is known for A, we show how to obtain the corresponding standard form of A(c). The web hyperlink matrix G(c) used by Google for computing the PageRank is a special case in which A is real, nonnegative, and row stochastic (taking into consideration the dangling nodes), c ∈ (0, 1), x is the vector of all ones, and v is a positive probability vector. The PageRank vector (the normalized dominant left eigenvector of G(c)) is therefore an explicit rational function of c. Extrapolation procedures on the complex field may give a practical and efficient way to compute the PageRank vector when c is close to 1.
4 citations