QR decomposition
Note that the singular values are real and right- and left- singular vectors are not required to form similarity transformations. One can iteratively alternate between the QR decomposition and the LQ decomposition to find the real diagonal Hermitian matrices. The QR decomposition gives and the LQ decomposition of gives . Thus, at every iteration, we have , update and repeat the orthogonalizations. Eventually, this iteration between QR decomposition and LQ decomposition produces left- and right- unitary singular matrices. This approach cannot readily be accelerated, as the QR algorithm can with spectral shifts or deflation. This is because the shift method is not easily defined without using similarity transformations. However, this iterative approach is very simple to implement, so is a good choice when speed does not matter. This method also provides insight into how purely orthogonal/unitary transformations can obtain the SVD.