Givens Rotation

A Givens rotation is an orthogonal matrix that acts as a rotation in the plane of two coordinate directions and leaves all other coordinates unchanged. One sign convention replaces the corresponding block of the identity matrix by

where . Its determinant is 1.

For real numbers and not both zero, taking , , and gives

Thus Givens rotations can introduce zeros into a matrix one at a time. Successive rotations give a QR decomposition and are particularly useful when only selected entries need to be eliminated. The Jacobi rotation matrix is a closely related construction with an angle chosen for matrix diagonalization.