Bulge chase preserves bidiagonal singular values

The first right rotation acts on the leading adjacent column pair of the active block and creates at most the first subdiagonal bulge. Each left rotation annihilates the current subdiagonal entry; because it mixes only two adjacent rows, it can create at most the next second-superdiagonal bulge. The following right rotation mixes only the adjacent columns supporting that bulge, annihilates it, and creates at most the next subdiagonal bulge. Induction along the block therefore keeps one bulge at a time and removes it at the next alternating rotation. The zero coupling at a deflated block endpoint prevents a bulge from leaving the block, so the final matrix is upper bidiagonal.

Every right operation multiplies on the right by a unitary Givens factor and every left operation multiplies on the left by the adjoint of a unitary Givens factor. Their products therefore give unitary $U_s,V_s$ and the final matrix is exactly $U_s^{\ast }B{V}_s$.

Left and right unitary factors preserve singular values, since $(U_s^{\ast }(B+E_D)V_s{)}^{\ast }(U_s^{\ast }(B+E_D)V_s)=V_s^{\ast }(B+E_D{)}^{\ast }(B+E_D)V_s$ is unitarily similar to $(B+E_D{)}^{\ast }(B+E_D)$. Deflation occurs before the sweep rotations, and setting each listed superdiagonal entry to zero adds exactly its negative there; hence the matrix entering the unitary sweep is $B+E_D$. With no deflation $E_D=0$, so step 1.2 gives equal singular values for $B_{+}$ and $B$.