On orthogonal block elimination
Christian H Bischof, Xiaobai Sun · 1996
. We consider the block elimination problem Q ` A 1 A 2 ' = ` \\GammaC 0 ' , where, given a matrix A 2 R m\\Thetak , A 11 2 R k\\Thetak , we try to find 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 \\Gamma 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 find Y 2 R m\\Thetar , S 2 R r\\Thetar , 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 \\GammaT , 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 th...