Givens Rotation Matrix Transformation Algorithm

Givens rotation is a fundamental orthogonal transformation method in numerical linear algebra, primarily used to zero out specific elements in a matrix while preserving its norm. The core concept involves constructing a 2D rotation matrix (Givens rotation matrix) that operates on two selected rows or columns of the matrix to achieve elimination.

Givens rotation matrix is a unitary orthogonal matrix typically applied to annihilate off-diagonal elements. For instance, in QR decomposition, Givens transformations progressively eliminate non-triangular elements to obtain an upper triangular matrix R. Compared to Householder transformations, Givens rotations are more suitable for sparse matrices or scenarios requiring localized elimination due to their targeted nature.

Key implementation steps include: Rotation angle calculation: Compute cosine and sine values based on target element coordinates using formulas like c = x_j/√(x_i² + x_j²) and s = -x_i/√(x_i² + x_j²) Givens matrix construction: Build a sparse rotation matrix affecting only two rows/columns, maintaining identity matrix structure elsewhere Transformation application: Perform left/right multiplication with the original matrix using efficient matrix update techniques

Givens transformations are widely employed in numerical computations including least squares problems, eigenvalue calculations, and matrix diagonalization. Their localized nature ensures controllable computational complexity, making them particularly valuable for high-performance computing and real-time systems where partial updates are required.