Tridiagonaliserde

Tridiagonaliserde refers to a matrix that has been transformed into a tridiagonal matrix through a process of similarity transformation. A tridiagonal matrix is a sparse matrix that has non-zero elements only on its main diagonal, the first superdiagonal, and the first subdiagonal. In other words, all other elements are zero.

The process of tridiagonalisation is important in numerical linear algebra. It is often a preliminary step in more complex matrix computations, such as finding eigenvalues and eigenvectors. By transforming a larger, denser matrix into a tridiagonal form, subsequent calculations can become significantly more efficient due to the reduced number of non-zero elements.

The transformation is achieved using a similarity transformation, which involves multiplying the original matrix A by an invertible matrix P and its inverse P⁻¹ such that the resulting matrix B = P⁻¹AP is tridiagonal. The eigenvalues of the original matrix A are preserved in the transformed tridiagonal matrix B. This preservation of eigenvalues is a key reason for employing tridiagonalisation. Various algorithms exist to perform this transformation, with the Householder reflection method being a common and stable approach. This method iteratively applies orthogonal transformations to zero out elements above and below the first subdiagonal.