In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a bidiagonal matrix is a
banded matrix
In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal ''band'', comprising the main diagonal and zero or more diagonals on either side.
Band matrix
Bandwidt ...
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:
:
and the following matrix is lower bidiagonal:
:
Usage
One variant of the
QR algorithm
In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a Matrix (mathematics), matrix. The QR algorithm was developed in the late 1950s by John ...
starts with reducing a general matrix into a bidiagonal one,
and the
singular value decomposition
In linear algebra, the singular value decomposition (SVD) is a Matrix decomposition, factorization of a real number, real or complex number, complex matrix (mathematics), matrix into a rotation, followed by a rescaling followed by another rota ...
(SVD) uses this method as well.
Bidiagonalization
Bidiagonalization allows guaranteed accuracy when using
floating-point arithmetic
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...
to compute singular values.
See also
*
List of matrices
A list is a set of discrete items of information collected and set forth in some format for utility, entertainment, or other purposes. A list may be memorialized in any number of ways, including existing only in the mind of the list-maker, but ...
*
LAPACK
LAPACK ("Linear Algebra Package") is a standard software library for numerical linear algebra. It provides routines for solving systems of linear equations and linear least squares, eigenvalue problems, and singular value decomposition. It als ...
*
Hessenberg form — The Hessenberg form is similar, but has more non-zero diagonal lines than 2.
References
*
External links
High performance algorithmsfor reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form
Linear algebra
Sparse matrices
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