Givensrotasjoner

Givens rotations are a fundamental concept in linear algebra and numerical analysis, primarily used to introduce zeros into specific positions within a matrix. They are a type of orthogonal transformation, meaning they preserve the Euclidean norm of vectors. A Givens rotation is defined by an angle and a plane, effectively rotating a vector within that plane.

In practice, a Givens rotation is represented by a matrix. For a 2D plane within an n-dimensional space, the Givens rotation matrix will be an n x n identity matrix with specific entries modified. The goal is typically to zero out a single element, often below the diagonal in a column. For example, to zero out the element at row i and column j of a matrix A, one would construct a Givens rotation matrix R such that RA has a zero at that position.

The key advantage of Givens rotations is their sparsity and the fact that they can be applied sequentially to transform a matrix into a desired form, such as upper triangular or bidiagonal. This makes them computationally efficient for certain algorithms. They are particularly useful in algorithms like QR decomposition and singular value decomposition (SVD), where controlled introduction of zeros is crucial for achieving the desired matrix factorization. Unlike Householder reflections, which zero out multiple elements in a column with a single reflection, Givens rotations are more localized in their effect.