In mathematics, a bidiagonal matrix is a banded matrix with non-zero entries along the main diagonal and either the diagonal above or the diagonal below. This means there are exactly two non-zero diagonals in the matrix. When the diagonal above the main diagonal has the non-zero entries the matrix is upper bidiagonal. When the diagonal below the main diagonal has the non-zero entries the matrix is lower bidiagonal. For example, the following matrix is upper bidiagonal: [math]\displaystyle{ \begin{pmatrix} 1 & 4 & 0 & 0 \\\ 0 & 4 & 1 & 0 \\\ 0 & 0 & 3 & 4 \\\ 0 & 0 & 0 & 3 \\\ \end{pmatrix} }[/math] and the following matrix is lower bidiagonal: [math]\displaystyle{ \begin{pmatrix} 1 & 0 & 0 & 0 \\\ 2 & 4 & 0 & 0 \\\ 0 & 3 & 3 & 0 \\\ 0 & 0 & 4 & 3 \\\ \end{pmatrix}. }[/math] ## Contents * 1 Usage * 1.1 Bidiagonalization * 2 See also * 3 References * 4 External links ## Usage One variant of the QR algorithm starts with reducing a general matrix into a bidiagonal one,[1] and the singular value decomposition (SVD) uses this method as well. ### Bidiagonalization Main page: Bidiagonalization Bidiagonalization allows guaranteed accuracy when using floating-point arithmetic to compute singular values.[2] ## See also * List of matrices * LAPACK * Hessenberg form – The Hessenberg form is similar, but has more non-zero diagonal lines than 2. ## References * Stewart, G. W. (2001) Matrix Algorithms, Volume II: Eigensystems. Society for Industrial and Applied Mathematics. ISBN:0-89871-503-2. 1. ↑ Bochkanov Sergey Anatolyevich. ALGLIB User Guide - General Matrix operations - Singular value decomposition . ALGLIB Project. 2010-12-11. URL:http://www.alglib.net/matrixops/general/svd.php. Accessed: 2010-12-11. (Archived by WebCite at https://www.webcitation.org/5utO4iSnR) 2. ↑ Fernando, K.V. (1 April 2007). "Computation of exact inertia and inclusions of eigenvalues (singular values) of tridiagonal (bidiagonal) matrices". Linear Algebra and Its Applications 422 (1): 77–99. doi:10.1016/j.laa.2006.09.008. ## External links * High performance algorithms for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form * v * t * e Matrix classes Explicitly constrained entries| * (0,1) * Alternant * Anti-diagonal * Anti-Hermitian * Anti-symmetric * Arrowhead * Band * Bidiagonal * Binary * Bisymmetric * Block-diagonal * Block * Block tridiagonal * Boolean * Cauchy * Centrosymmetric * Conference * Complex Hadamard * Copositive * Diagonally dominant * Diagonal * Discrete Fourier Transform * Elementary * Equivalent * Frobenius * Generalized permutation * Hadamard * Hankel * Hermitian * Hessenberg * Hollow * Integer * Logical * Markov * Metzler * Monomial * Moore * Nonnegative * Partitioned * Parisi * Pentadiagonal * Permutation * Persymmetric * Polynomial * Positive * Quaternionic * Sign * Signature * Skew-Hermitian * Skew-symmetric * Skyline * Sparse * Sylvester * Symmetric * Toeplitz * Triangular * Tridiagonal * Unitary * Vandermonde * Walsh * Z Constant| * Exchange * Hilbert * Identity * Lehmer * Of ones * Pascal * Pauli * Redheffer * Shift * Zero Conditions on eigenvalues or eigenvectors| * Companion * Convergent * Defective * Diagonalizable * Hurwitz * Positive-definite * Stability * Stieltjes Satisfying conditions on products or inverses| * Congruent * Idempotent or Projection * Invertible * Involutory * Nilpotent * Normal * Orthogonal * Orthonormal * Singular * Unimodular * Unipotent * Totally unimodular * Weighing With specific applications| * Adjugate * Alternating sign * Augmented * Bézout * Carleman * Cartan * Circulant * Cofactor * Commutation * Confusion * Coxeter * Derogatory * Distance * Duplication * Elimination * Euclidean distance * Fundamental (linear differential equation) * Generator * Gramian * Hessian * Householder * Jacobian * Moment * Payoff * Pick * Random * Rotation * Seifert * Shear * Similarity * Symplectic * Totally positive * Transformation * Wedderburn * X–Y–Z Used in statistics| * Bernoulli * Centering * Correlation * Covariance * Design * Dispersion * Doubly stochastic * Fisher information * Hat * Precision * Stochastic * Transition Used in graph theory| * Adjacency * Biadjacency * Degree * Edmonds * Incidence * Laplacian * Seidel adjacency * Skew-adjacency * Tutte Used in science and engineering| * Cabibbo–Kobayashi–Maskawa * Density * Fundamental (computer vision) * Fuzzy associative * Gamma * Gell-Mann * Hamiltonian * Irregular * Overlap * S * State transition * Substitution * Z (chemistry) Related terms| * Jordan canonical form * Linear independence * Matrix exponential * Matrix representation of conic sections * Perfect matrix * Pseudoinverse * Quaternionic matrix * Row echelon form * Wronskian * List of matrices * Category:Matrices 0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Bidiagonal matrix. Read more | Retrieved from "https://handwiki.org/wiki/index.php?title=Bidiagonal_matrix&oldid=2643604" *[v]: View this template *[t]: Discuss this template *[e]: Edit this template