Thus the lower formula_10 rows are zero. Since formula_11, it follows that these rows in "C"("i") in this basis contain a right block which is a formula_12 unit matrix, with zeros in the rest of these rows. But since "C"("i") has rank "N" − "r""i", it must be zero elsewhere. Thus it is diagonal in this basis as well. It follows that all the non-zero eigenvalues of both "B"("i") and "C"("i") are +1. Moreover, the above analysis can be repeated in the diagonal basis for formula_13. In this basis formula_14 is the identity of an formula_15 vector space, so it follows that both "B"(2) and formula_16 are simultaneously diagonalizable in this vector space (and hence also together with "B"(1)). By iteration it follows that all "B"-s are simultaneously diagonalizable.