On the reduction of an arbitrary real square matrix to tridiagonal form
H. H. Wang, Robert Todd Gregory · Mathematics of Computation · 1964
3. La Budde [1, p. 436] says, In order to continue the algorithm into step j + 1, we must be certain that S+ 1 0 ... Now a, b are arbitrary (except for sign) so we may theoretically choose a, b I so thatS11 $0.... l al, b I could be determined by trial and error starting . It is our purpose to demonstrate that we may not always be able to choose I a i and I b I in order to insure that S+ , $0. We shall do this by displaying some matrices which are counter examples. These matrices fall into two categories. First category. If at some step of the reduction, say the ith step, the matrix A is in the reduced form (here and afterwards the subscripts denote the size of the submatrix)