Bidiagonal matrix

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:

{\begin{pmatrix}1&4&0&0\\0&4&1&0\\0&0&3&4\\0&0&0&3\\\end{pmatrix}}

and the following matrix is lower bidiagonal:

{\begin{pmatrix}1&0&0&0\\2&4&0&0\\0&3&3&0\\0&0&4&3\\\end{pmatrix}}.

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

Bidiagonalization allows guaranteed accuracy when using floating-point arithmetic to compute singular values.^[2]

References

Stewart, G.W. (2001). Eigensystems. Matrix Algorithms. Vol. 2. Society for Industrial and Applied Mathematics. ISBN 0-89871-503-2.

^ Anatolyevich, Bochkanov Sergey (2010-12-11). "Matrix operations and decompositions — Other operations on general matrices — SVD decomposition". ALGLIB User Guide, ALGLIB Project. Accessed: 2010-12-11. (Archived by WebCite at)
^ 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. S2CID 122729700.

External links

High performance algorithms for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form

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[1] Anatolyevich, Bochkanov Sergey (2010-12-11). "Matrix operations and decompositions — Other operations on general matrices — SVD decomposition". ALGLIB User Guide, ALGLIB Project. Accessed: 2010-12-11. (Archived by WebCite at)

[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. S2CID 122729700.

[1]

[2]

v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal 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 Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel 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 normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
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Usage

Bidiagonalization

See also

References

External links