Showing papers in "Linear Algebra and its Applications in 2001"

TL;DR: In this article, the concept of a Leonard system was introduced, and it was shown that for any Leonard pair A,A* on V, there exists a sequence of scalars β,γ,γ*,ϱ,ϱ* taken from K such that both

TL;DR: In this paper, the authors review and compare the existing perturbation bounds for the stationary distribution of a finite, irreducible, homogeneous Markov chain, and compare them with the perturbations of a fixed stationary distribution.

TL;DR: In this paper, a new family of indecomposable positive linear maps based on entangled quantum states is introduced, and the notion of an unextendible product basis is introduced.

TL;DR: In this paper, the authors give a detailed proof for two discrete analogues of Courant's Nodal Domain Theorem (CNDT) and show that they are equivalent.

TL;DR: In this paper, additive perturbation results for Drazin inverses are given, where one of the products of these matrices vanishes and some special applications of this are also considered.

TL;DR: In this paper, explicit inverses of tridiagonal 2-Toeplitz and 3-Toplitz matrices were given, which generalize some well-known results concerning the inverse of a 2-to-toplitz matrix.

TL;DR: A much shorter and more elegant, but still sharp in decisive quantities, convergence rate estimate of the same method that also holds for a generalized symmetric definite eigenvalue problem is discovered and proved.

TL;DR: In this paper, the concept of left and right eigenvalues for a quaternionic matrix was introduced, and the properties, quantities and relationship of these eigenvectors were investigated.

TL;DR: In this article, a corrigendum/addendum supplies corrected statements and proofs of some results in a paper appearing in Linear Algebra Appl. 161 (1992) 227-263.

TL;DR: In this paper, explicit formulae for the elements of the inverse of a general tridiagonal matrix are presented by first extending results on the explicit solution of a second-order linear homogeneous difference equation with variable coefficients to the nonhomogeneous case, and then applying these extended results to a boundary value problem.

TL;DR: The given theory provides sharp convergence estimates for the eigenvalue approximations showing that multigrid eigen value/vector computations can be done with comparable efficiency as known from multigrids methods for boundary value problems.

TL;DR: An algorithm for computing partial Pade approximants via suitable rank-1 updates of the tridiagonal matrices generated by the Lanczos process is presented.

TL;DR: In this paper, a version of Fejer-Riesz factorization and factorization of positive operator-valued polynomials in several non-commuting variables is presented.

TL;DR: A method is described for the computation of rigorous error bounds for multiple or nearly multiple eigenvalues, and for a basis of the corresponding invariant subspaces, based on a quadratically convergent Newton-like method.

TL;DR: In this article, the authors propose a new approach based on a suitable fixed-point formulation of the problem and uses, as an essential ingredient, norm bounds for the inverse of the linearization of the given problem at some approximate solution ω which is computed numerically.

TL;DR: In this article, an incomplete LU decomposition with pivoting is presented that progressively monitors the growth of the inverse factors of L, U and is used as feedback for dropping entries in L and U.

TL;DR: The problem of reconstructing a discrete two-dimensional set from its two orthogonal projection (H,V) when the set satisfies some convexity conditions can be solved in polynomial time also in a class of discrete sets which is larger than the class of convex polyominoes, namely, in theclass of 8-connected convex sets.

TL;DR: In this paper, the extremal graphs for which equality in de Caen's inequality holds and then apply the inequality to give an upper bound for the largest Laplacian eigenvalue λ 1 (G) of a graph.

TL;DR: In this article, rank equalities for idempotent and involutary matrices are established, including the rank of the difference, the sum, the product and the commutator.

TL;DR: In this paper, upper and lower bounds for the spread of the adjacency matrix of a simple graph were obtained for the eigenvalues λ 1 λ 2 ··· λn.

TL;DR: In this paper, the nonlinear matrix equation X + A*X(-1)A = P, where A, P are n x n complex matrices with P Hermitian positive definite, and A* denotes the conjugate transpose of a matrix A.

TL;DR: In this article, it was shown that a sequence of Toeplitz matrices is spectrally distributed in the sense of the singular values as the measurable function θ = ∑ α ∏ β a αβ for a α β ∈ L 1 and α and β ranging in any finite set of values.

TL;DR: In this article, sufficient conditions for the existence of positive definite solutions of two different solutions of the equation X + A ∗ X −2 A = I were derived, and Numerical experiments were discussed.

TL;DR: In this article, Jacobi et al. developed a quaternion characterization of the 4×4 symplectic orthogonal group and the subspaces associated with a matrix in any one of the four classes under consideration.

TL;DR: In this article, a general class of rational matrix equations, which contains the continuous (CARE) and discrete (DARE) algebraic Riccati equations as special cases, are studied.

TL;DR: In this article, the singular value distribution of Gaussian random matrices (i.e., GE) of size N is derived directly from the SVD form, which also takes advantage of the rotational invariance of GE and the Lie algebra of the orthogonal group.

TL;DR: In this paper, the quadratic numerical range (QR) was studied for 2×2 -block operator matrices and the main results were a spectral inclusion theorem, an estimate of the resolvent in terms of the QR, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of corresponding Riccati equations and a block diagonalization.

TL;DR: In this paper, the Stirling numbers of the first kind s(n,k) and of the second kind n,k were studied, and it was shown that these matrices can be factorized by the Pascal matrices, and the LDU-factorization of a Vandermonde matrix of the form V n (x,x+1,…,x +n−1) for any real number x is obtained.

TL;DR: In this article, the authors considered bounded families of complex n n-matrices and investigated how the norm of products of matrices behaves as the number of factors goes to infinity.

TL;DR: In this article, it was shown that if σ(A)∩(−∞,0]=∅, then A has a unique square root B∈M n (C ) with σ (B) in the open right (complex) half plane.