Unit 11.4.2 Casting the reduction to bidiagonal form in terms of matrix-matrix multiplication
As was discussed in Unit 10.4.3 for the reduction to tridiagonal form, reduction to bidiagonal form can only be partly cast in terms of matrix-matrix multiplication. As for the reduction to tridiagonal form, we recommend
 Field G. Van Zee, Robert A. van de Geijn, Gregorio Quintana-Ortí, G. Joseph Elizondo, Families of Algorithms for Reducing a Matrix to Condensed Form, ACM Transactions on Mathematical Software (TOMS) , Vol. 39, No. 1, 2012.
Bidiagonal, tridiagonal, and upper Hessenberg form are together referred to as condensed form.