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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''':
:<math>\begin{pmatrix}
1 & 4 & 0 & 0 \\
0 & 4 & 1 & 0 \\
0 & 0 & 3 & 4 \\
0 & 0 & 0 & 3 \\
\end{pmatrix}</math>

and the following matrix is '''lower bidiagonal''':

:<math>\begin{pmatrix}
1 & 0 & 0 & 0 \\
2 & 4 & 0 & 0 \\
0 & 3 & 3 & 0 \\
0 & 0 & 4 & 3 \\
\end{pmatrix}.</math>

==Usage==
One variant of the [[QR algorithm]] starts with reducing a general matrix into a bidiagonal one,<ref>{{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)</ref>
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.<ref>{{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 }}</ref>

{{expand Section|date=January 2017}}

==See also==
* [[List of matrices]]
* [[LAPACK]]
* [[Hessenberg form]] — The Hessenberg form is similar, but has more non-zero diagonal lines than 2.

==References==
{{refbegin}}
* {{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 }}
{{refend}}
{{Reflist}}

==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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