Matrix Computations and Semiseparable Matrices: Eigenvalue and Singular Value Methods

Notation

Notation Throughout the book we use the following notation, unless stated otherwise. x, y, z . . . . . . . . . Denotes the vector space spanned by the vectors x, y and z. A = (ai,j)i,j . . . . . . A general matrix A with elements ai,j. A(i : j, k : l). . . . . . Submatrix of A, consisting of rows i up to and including j, and columns k up to and including l. A(α; β) . . . . . . . . . . Submatrix of A, consisting of indices out of the set α and β. α × β . . . . . . . . . . . . α × β denotes the product set {(i, j)|i ∈ α, j ∈ β}, with α and β sets. C . . . . . . . . . . . . . . . Denotes the set of complex numbers. Cn[z] . . . . . . . . . . . . Denotes the set of complex polynomials of degree ≤ n. cond (A) . . . . . . . . The condition number of a matrix A. d(S, T ) . . . . . . . . . . Distance between the subspaces S and T . D. . . . . . . . . . . . . . . . A diagonal matrix. deg (p) . . . . . . . . . . The degree of a polynomial p. det (A) . . . . . . . . . . The determinant of a matrix A. diag (d) . . . . . . . . . . Denotes a diagonal matrix, with as diagonal elements, the elements from the vector d. ei . . . . . . . . . . . . . . . The i-th basis vector. G . . . . . . . . . . . . . . . . A Givens transformation.·H . . . . . . . . . . . . . . . The hermitian conjugate of the involved matrix or vector. H . . . . . . . . . . . . . . . . A (generalized) Hessenberg matrix. H∞×∞ = [hj,k]. . . An infinite Hessenberg matrix with elements hjk. Ik . . . . . . . . . . . . . . . The identity matrix of size k × k. κ(A) . . . . . . . . . . . . . The condition number of the matrix A. Kk(A, v) . . . . . . . . . The Krylov subspace of the matrix A, with vector v of dimension k. Λi = {λ1, . . . , λn} The λi denote the eigenvalues and Λ the spectrum of a matrix. μ . . . . . . . . . . . . . . . . The shift in the QR-method. p̂j:i(λ) . . . . . . . . . . Equals pj(λ)pj−1(λ) . . . pi(λ). p0:∞ = [p0, p1, . . .] An infinite vector with elements pi. xv xvi Notation Pn . . . . . . . . . . . . . . . . . Denotes the set of polynomials of degree less than n. PM n . . . . . . . . . . . . . . . . . Denotes the monic set of polynomials of degree less than n. Range(A) . . . . . . . . . Denotes the vector space spanned by the columns of A. R . . . . . . . . . . . . . . . . . . Denotes the set of real numbers. Rn[z]. . . . . . . . . . . . . . . Denotes the polynomials in R of degree ≤ n. rank (A) . . . . . . . . . . . The rank of a matrix A. Σ = {σ1, . . . , σn} . . . The singular values of a given matrix. S. . . . . . . . . . . . . . . . . . . A subspace S. S . . . . . . . . . . . . . . . . . . A semiseparable matrix. S(u, v). . . . . . . . . . . . . A symmetric generator representable semiseparable matrix with generators u and v.·T . . . . . . . . . . . . . . . . . . The transpose of the involved matrix or vector. T . . . . . . . . . . . . . . . . . . A tridiagonal matrix. T . . . . . . . . . . . . . . . . . . Denotes the unit circle. tril(A, p) . . . . . . . . . . Denotes the lower triangular part of the matrix A, below and including subdiagonal p. triu(A, p). . . . . . . . . . . Denotes the upper triangular part of the matrix A, above and including superdiagonal p. Q . . . . . . . . . . . . . . . . . . A unitary(orthogonal) matrix (with regard to the QRfactorization ). R . . . . . . . . . . . . . . . . . . An upper triangular matrix (with regard to the QRfactorization ). uT = [u1, u2, . . . , un] A column vector u, with entries ui. All vectors are column vectors and denoted in bold. u(i : j) = ui:j . . . . . . A subvector of u, consisting of the elements i up to and including j. Z . . . . . . . . . . . . . . . . . . A Hessenberg-like matrix. z̄ . . . . . . . . . . . . . . . . . . . The complex conjugate of z. ...