Algorithms for Matrix Canonical Forms by Arne Storjohann

By Arne Storjohann

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For example, the nonzero rows of H are the canonical “Hermite basis” for A. 4. The first r rows of any echelon form of A are a basis for A. A basis G for A is sometimes called a (right) matrix gcd. 5. If r = m, and G is a basis for A, then all entries in AGadj are divisible by det G, and L(AGadj (1/ det G)) = Rm . Consider for the moment that n = 2m, r = m, and A can be written as A1 A2 where both A1 and A2 are square nonsingular. Let E ∈ Rr×n be such that EA = G. Then E is called a solution to the extended matrix gcd problem.

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= . where Ki is a kernel for Hi , i = 1, 2. If K1 F = SH2 then K is a kernel for H. If in addition H1 and H2 are in weak Howell form and H1 has no zero row, then H is in weak Howell form. 2. 2  d1    ,  I Ik    c2 Moreover, there exists a subrouting Combine that takes as input (Q1 , U1 , C1 , Q2 , U2 , C2 ), produces as output (Q, U, C), and has cost bounded by O(n max(r1 , r2 ) min(r1 , r2 )θ−1 ) arithmetic operations.

3. By carefully specifying the number of rows to operations for the bottom-to-top method. A subtlety we face is that r may not unique with respect to A but only with respect to the exact method used to compute T . ) We deal with this problem by ensuring that, when applying the bottom-to-top method, the number of nonzero rows the succesive echelon forms we compute is nondecreasing. Furthermore, if we want a minimal echelon form then we ensure that each echelon form computed is minimal. 58 CHAPTER 3.

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