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By Arne Storjohann

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Extra resources for Algorithms for Matrix Canonical Forms

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Let A ∈ Rn×m and k ∈ N satisfy 0 ≤ k ≤ n and n − k ≥ r where r is the number of nonzero rows in the Howell form of A. 3) with U unimodular, A¯ the first k rows of A, H a weak Howell basis for A, W a kernel for T , K a kernel for H and S such that A¯ = SH. 5. 4 is correct. 4 (Weak Howell) is given on page 76. are all in Rn×n and the block decomposition is conformal, then where (Q, U, C) =  Ik  I r1   Ir2 q1 q2 I = I r1 Ir2   c12   d2  I Proof. Assume for now that R is a stable ring.

Let A ∈ Rn×m have full column rank and let d ∈ R ¯ is a Hermite form of φd2 (A) over satisfy det L(A)|d, d ∈ R \ R∗ . If H 2 −1 ¯ R/(d ), then φ (H) is the Hermite form of A over R. ¯ Then H is in Hermite form over R. We need Proof. Let H = φ−1 (H). to show that H is the Hermite form of A. 11 A ∼ = H d2 I . 2) Thus H is left multiple of A. 3 it will suffice to show that det L(A) = det L(H). 1. 2). All such minors which involve a row from d2 I will be a multiple of d2 . We may deduce that H has rank m and that (d2 , det L(H)) = (det L(A)).

5 become O(nmrθ−2 (log β) + nm(log r) B(log β)) and O(nrθ−1 (log r)(log β)) word operations where β = rN . Proof. Perform the following: T := a copy of A; z := 1; r := 1; for i from 1 do d := min(max(r, 1), n − z); # Let (Ti , ri , zi , di ) be a copy of (T, r, z, d) at this point. z := z + d; B := the last z rows of T ; V := a principal left transform such that V B is in echelon form; In−z Ui := ; V T := U T ; r := the number of nonzero rows in V B; if z = n then break fi; od; Induction on i shows that     Ui I ∗ I  Ti ∗  ∗   ¯ ∗  Ti+1 ∗   ¯  = ∗     62 CHAPTER 3.

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