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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**:

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

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

Bidiagonalization allows guaranteed accuracy when using floating-point arithmetic to compute singular values.^{[2]}

- List of matrices
- LAPACK
- Hessenberg form — The Hessenberg form is similar, but has more non-zero diagonal lines than 2.

- 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.

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