On Orthogonal Block Elimination

We consider the block elimination problem Q A 1 A 2 = ?C 0 , where, given a matrix A 2 R mk , A 11 2 R kk , we try to nd a matrix C with C T C = A T A and an orthogonal matrix Q that eliminates A 2. Sun and Bischof recently showed that any orthogonal matrix can be represented in the so-called basis-kernel representation Q = Q(Y; S) = I ? Y ST T. Applying this framework to the block elimination problem, we show that there is considerable freedom in solving the block elimination problem and that, depending on A and C, we can nd Y 2 R mr , S 2 R rr , where r is between rank(A 2) and k, to solve the block elimination problem. We then introduce the canonical basis Y = A 1 + C A 2 and the canonical kernel S = (A 1 + C) y C ?T , which can be determined easily once C has been computed, and relate this view to previously suggested approaches for computing block orthogonal matrices. We also show that the condition of S has a fundamental impact on the numerical stability with which Q can be computed and prove that the well-known compact WY representation approach, employed, for example, in LAPACK, leads to a well-conditioned kernel. Lastly, we discuss the computational promise of the canonical basis and kernel, in particular in the sparse setting, and suggest pre-and postconditioning strategies to ensure that S can be computed reliably and is wellconditioned. in the directory pub/prism.