GivensRotationen

Givens rotations are a method for introducing zeros into a matrix in a controlled manner. They are a specific type of orthogonal transformation. A Givens rotation in the plane is defined by a 2x2 matrix of the form:

[ cos(theta) sin(theta) ]

[ -sin(theta) cos(theta) ]

or

[ cos(theta) -sin(theta) ]

[ sin(theta) cos(theta) ]

where theta is the angle of rotation. The key idea is to apply this 2x2 rotation to two specific rows (or columns) of a larger matrix to zero out a particular element. The cosine and sine values are chosen such that they precisely cancel out the element targeted for elimination.

In practice, the angle theta is not explicitly computed. Instead, the cosine and sine values are determined directly from the elements of the matrix being operated upon. If we want to zero out the element a_ij in a vector, and the element at the same position in another vector is a_kj, we can compute c = a_ij / sqrt(a_ij^2 + a_kj^2) and s = a_kj / sqrt(a_ij^2 + a_kj^2). Applying the rotation matrix [ c s ; -s c ] to the vector [ a_ij ; a_kj ] results in [ sqrt(a_ij^2 + a_kj^2) ; 0 ].

Givens rotations are particularly useful in algorithms like the QR decomposition, where they are applied sequentially to zero out elements below the diagonal of a matrix. This process is stable and numerically robust, making it a preferred method in many scientific and engineering applications, especially when dealing with sparse matrices. They are also used in eigenvalue computations and solving linear systems.