Bidiagonalization
Encyclopedia
Bidiagonalization is one of unitary (orthogonal) matrix decomposition
s such that U* A V = B, where U and V are unitary (orthogonal
) matrices; * denotes Hermitian transpose; and B is upper bidiagonal
. A is allowed to be rectangular.
For dense matrix, the left and right unitary matrices are obtained by means of Householder reflections, also known as Golub-Kahan bidiagonalization. For large matrix, they are calculated iteratively by using Lanczos method
, referred to as Golub-Kahan-Lanczos method.
Bidiagonalization has a very similar structure to the singular value decomposition
(SVD). However, it is computed within finite operations, while SVD requires iterative schemes to find singular values. It is because the singular values are roots of characteristic polynomial
s of A* A, where A is assumed to be tall.
Matrix decomposition
In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.- Example :...
s such that U* A V = B, where U and V are unitary (orthogonal
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
) matrices; * denotes Hermitian transpose; and B is upper bidiagonal
Bidiagonal matrix
A bidiagonal matrix is a matrix with non-zero entries along the main diagonal and either the diagonal above or the diagonal below.So that means there are two non zero diagonal in the matrix....
. A is allowed to be rectangular.
For dense matrix, the left and right unitary matrices are obtained by means of Householder reflections, also known as Golub-Kahan bidiagonalization. For large matrix, they are calculated iteratively by using Lanczos method
Lanczos algorithm
The Lanczos algorithm is an iterative algorithm invented by Cornelius Lanczos that is an adaptation of power methods to find eigenvalues and eigenvectors of a square matrix or the singular value decomposition of a rectangular matrix. It is particularly useful for finding decompositions of very...
, referred to as Golub-Kahan-Lanczos method.
Bidiagonalization has a very similar structure to the singular value decomposition
Singular value decomposition
In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
(SVD). However, it is computed within finite operations, while SVD requires iterative schemes to find singular values. It is because the singular values are roots of characteristic polynomial
Characteristic polynomial
In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
s of A* A, where A is assumed to be tall.