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By Arne Storjohann

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7. Let R = ZN . 6 becomes O(nmrθ−2 (log β) + nm(log r) B(log β)) word operations where β = rN . 2. 8. Let A ∈ Rn×m . If either R is a stable ring or A has at least first r rows zero (where r is the number of rows in the Howell basis of A) then a Howell transform for A can be computed in O(nmrθ−2 ) basic operations of type {Arith, Gcdex, Ann, Quo, Stab}. Proof. Compute an index weak Howell transform (Q, U, C, W, r) for (A, 0). Now recover an upper triangular and unimodular R= ∗ I ¯ CA is in Hermite (and Howell) form.

A straightforward multiplication verifies that Q2 U2 ((C2 − I)W1 + I)Q1 U1 C1 A = T . Then T is left equivalent to A hence H is a weak Howell basis also for A. This also shows that r1 + r2 = r. Now assume that the input matrix A has rows k + 1, k + 2, . . , k + r zero. Using induction we can show that all subproblems will satisfy the same condition and hence A[k + 1, 0] will also be zero. In this case no operations of type Stab will be required. 6. Let A ∈ Rn×m and k be such that (A, k) is valid input to algorithm WeakHowell.

Let A ∈ Rn×m . An echelon form T of A together with a principal left transform U such that U A = T can be recovered in O(mnθ−1 ) basic operations of type {Arith, Gcdex}. The user may choose to have either (but not necessarily both) of the following conditions satisfied: 60 CHAPTER 3. 1. TRANSFORMATION TO ECHELON FORM 61 • (a) T will have at least k nonzero rows where k is maximal such that the last k rows of A are nonzero and in echelon form. 3. • (b) T will be a minimal echelon form of A. 5. Let A ∈ Rn×m .

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Algorithms for Matrix Canonical Forms by Arne Storjohann

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