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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,{{cite web |first=Bochkanov Sergey |last=Anatolyevich |title=Matrix operations and decompositions — Other operations on general matrices — SVD decomposition |date=2010-12-11 |work=ALGLIB User Guide, ALGLIB Project |url=https://www.alglib.net/matrixops/general/svd.php}} Accessed: 2010-12-11. (Archived by WebCite at)

and the singular value decomposition (SVD) uses this method as well.

=Bidiagonalization=

{{Main|Bidiagonalization}}

Bidiagonalization allows guaranteed accuracy when using floating-point arithmetic to compute singular values.{{cite journal |last1=Fernando |first1=K.V. |title=Computation of exact inertia and inclusions of eigenvalues (singular values) of tridiagonal (bidiagonal) matrices |journal=Linear Algebra and Its Applications |date=1 April 2007 |volume=422 |issue=1 |pages=77–99 |doi=10.1016/j.laa.2006.09.008 |s2cid=122729700 |ref=inertia|doi-access=free }}

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See also

References

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  • {{cite book |first=G.W. |last=Stewart |title=Eigensystems |series=Matrix Algorithms |volume=2 |publisher=Society for Industrial and Applied Mathematics |location= |date=2001 |isbn=0-89871-503-2 }}

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External links

  • [http://www.cs.utexas.edu/users/flame/pubs/flawn53.pdf High performance algorithms] for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form

{{Matrix classes}}

Category:Linear algebra

Category:Sparse matrices

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