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Bidiagonal matrix

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Inmathematics, abidiagonal matrix is abanded matrix with non-zero entries along the main diagonal andeither 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 isupper bidiagonal. When the diagonal below the main diagonal has the non-zero entries the matrix islower bidiagonal.

For example, the following matrix isupper 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 islower bidiagonal:

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

The eigenvalues of a bidiagonal matrix (of either type) are given by the entries of the diagonal.

Usage

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One variant of theQR algorithm starts with reducing a general matrix into a bidiagonal one,^[1]and thesingular value decomposition (SVD) uses this method as well.

Bidiagonalization

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Main article:Bidiagonalization

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

This sectionneeds expansion. You can help byadding missing information.(January 2017)

References

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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

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

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