bidiagonalisaation

Bidiagonalization is a process in numerical linear algebra used to transform a given matrix into a bidiagonal matrix, which is a matrix with non-zero elements only on its main diagonal and its first superdiagonal or subdiagonal. This transformation is typically achieved through a sequence of orthogonal similarity transformations. The primary goal of bidiagonalization is to simplify the matrix for further computations, particularly in algorithms like the singular value decomposition (SVD).

A common method for bidiagonalizing a real matrix A is the Golub-Kahan-Reinsch algorithm, which involves applying Householder reflections. First, a series of transformations are applied to zero out the elements below the first subdiagonal, resulting in an upper bidiagonal matrix. Subsequently, another series of transformations are applied to zero out the elements above the first superdiagonal, yielding a lower bidiagonal matrix. These transformations preserve the eigenvalues and singular values of the original matrix, making it a useful preliminary step for eigenvalue or SVD computations. The bidiagonal form is simpler to work with due to its sparse structure, enabling more efficient algorithms for tasks such as finding singular values.