A method and a system that multiplies a beamforming matrix by a unitary matrix that does not change the subspace of the beamforming matrix for form a converted matrix having a lower left triangle of zeros. A first column vector having a fewest number of elements of the converted matrix is quantized using a codebook and represented by a first codebook index. A Householder matrix is determined from the quantized first column vector and the converted matrix is multiplied by the Householder matrix. Quantizing, determining a Householder matrix from further column vectors of the converted matrix using a codebook and representing each respective column vector by further corresponding codebook index, and multiplying the converted matrix on the left by the determined Householder matrix for each respective column vector are recursively repeated. The first codebook index and further codebook indices are transmitted to a remote station for use in beamforming.