By Arne Storjohann

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By permuting the blocks T1 and T2 (if necessary) we may assume that T2 has at least as many rows as T1 . 3 to compute a principal left transform U2 such that U2 ∗ T1 ∗ 0 I ∗ T2 = T ∗ I 0 where T is in echelon form. 3b. 3a. In either case, T will have at least max(r(A1 ), r(A2 )) rows. This shows that fm,r (n) ≤ 2fm,r (n/2) + O((n + m)rθ−1 ). This resolves to fm,r (n) ≤ n/¯ rfm,r (¯ r) + O(nrθ−1 (log 2n/r) + nmrθ−2 ) where r¯ is the smallest power of two such that r¯ ≥ r.

Buchmann and Neis call the first r rows of H a standardized generating set for S(A). This chapter is joint work with Thom Mulders. An earlier version appears in (Mulders and Storjohann, 1998). . 1 CHAPTER 4. 2. THE HOWELL TRANSFORM Preliminaries 73 with c1 − d¯1 c2 d1 − d¯1 d2 Let A ∈ Rn×m . We say A is in weak Howell form if A satisfies (r1) and (r3) but not necessarily (r2). 1. Let u21 = u12 = u2 (¯ q1 + d2 q1 + c2 w1 + c¯2 w ¯1 )u1 ¯ u1 d1 u22 = u2 + u21 d¯1 H= H1 F H2 and −S K2 K1 K= .

Let A1 be the first m columns of A, and let 50 CHAPTER 2. ECHELON FORMS OVER FIELDS (U, P, r, h, d) be a fraction-free Gauss transform for (A1 , d0 ). Let F1 be the adjoint transform of (B1 , d0 ) where B1 is the principal min(m, r+1)th submatrix of A. 1) The principal min(m, r + 1)th submatrix of F equals F1 . 2) If P = In then the principal min(m, r + 1)th submatrix of U equals F1 . 3) If r < m then the last n − r − 1 rows of F are zero. Assume henceforth that r = m. Let F2 be the adjoint transform of (B2 , d) where B2 is the trailing (n − m) × (n − m) submatrix of U P A.