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New PDF release: Algorithms for Matrix Canonical Forms

By Arne Storjohann

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For 0 ≤ i ≤ r let H[i, ji ] be the first nonzero entry in row i. 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.

Jr ), and • H is the Hermite form of A. 77 78 CHAPTER 5. ECHELON FORMS OVER PIDS ¯ denote the fraction field of R. Note that for nonzero N ∈ R, the Let R residue class ring R/(N ) will be a PIR but not, in general, a PID. Recall the relationship between these rings in Figure 5. 1: Relationship between rings the relationship between these rings from the point of view of recovering various matrix invariants over R. The concrete rings which will most ¯ = Q and N is composite) and R = Q[x] interest us are R = Z (whereby R ¯ = Q(x) and N ∈ Q[x] has a nontrivial factorization).

The result follows. 2 The Index Reduction Transform Let A ∈ Rn×n be upper triangular with diagonal entries nonzero and in A(R). The motivating application of the algorithm developed in this 64 CHAPTER 3. TRIANGULARIZATON OVER RINGS section to compute a unit upper triangular U ∈ Rn×n such that U A is in Hermite form. For 1 ≤ i < j ≤ n, let φi,j (a) be a prescribed function on R which satisfies φi,j (a + φi,j (a)A[j, j]) = 0 for all a ∈ R. Different applications of the index reduction transform will call for different choices of the φ∗,∗ .

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Algorithms for Matrix Canonical Forms by Arne Storjohann

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