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Then 0 = j0 < j1 < j2 < . . < jr . (r2) H[i, ji ] ∈ A(R) and H[k, ji ] ∈ R(R, H[i, ji ]) for 1 ≤ k < i ≤ r. (r3) H has a minimal number of nonzero rows. Using these conditions we can distinguish between four forms as in Table 3. This chapter gives algorithms to compute each of these forms. 1: Non canonical echelon forms over a PIR include the time to recover a unimodular transform matrix U ∈ Rn×m such that U A = H. Some of our effort is devoted to expelling some or all of the logorithmic factors in the cost estimates in case a complete transform matrix is not required.

This condition on the determinant of U means that there also exists a V ∈ Rn×n with det V ⊥ d and V B ≡ A mod d. 11. Let A, B ∈ Rn×m . If A ∼ =d B then A dI ∼ = B dI . The next two observations captures the essential subtlety involved in computing the Hermite form “modulo d”. 12. Let a, d, h ∈ R be such that a|d and h only has prime divisors which are also divisors of d. If (d2 , h) = (a) then (h) = (a). 13. Let a, d, h ∈ R be such that a|d and h only has prime divisors which are also divisors of d.

A reduction transform for A ∈ Zn×n with respect to φ∗,∗ can be computed in O(nθ ) basic operations of type Arith plus fewer than nm calls to φ∗,∗ . 11. Let R = Z/(N ). 10 becomes O(nθ (log β) + n2 (log n) B(log β)) word operations where β = nN . Proof. Computing an index k reduction transform for an n × 1 matrix requires n − k < n applications of φ∗,∗ . Let fn (k) be the number of basic operations (not counting applications of φ∗,∗ ) required to compute an index k transform for an m × m matrix A where m ≤ n.

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Algorithms for Matrix Canonical Forms by Arne Storjohann

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