Another orthogonal matrix, revisited

A 2006 paper by Parlett and Barszcz [4] proposed the following problem: Given an unit vector q, compute the orthogonal Hessenberg matrix A with first column q. In complex form, this translates to completing the unitary Hessenberg matrix U with first column q. Looking at the matrix I−qq⁎ in the special form I−qq⁎=LD2L⁎, where L is n×(n−1) and lower triangular with 1's on its main diagonal and D2=diag(μ12,…,μn−12) is positive definite, Parlett observed that for i>j the entries of L˜=LD can be written as l˜ij=−qiq‾jμj/ρj, where q‾j denotes the complex conjugate of qj and ρi=∑j=i+1n|qj|2, ρn=0, and μi=ρi/ρi−1, for i=1,…,n. Furthermore, one solution to this problem is U=[qL˜]. Section 1 provides some background, as well as details on the derivation of Parlett's formula. Section 2 contains the main result, where Parlett's method is extended to “tall thin” matrices as suggested by Parlett in his original paper. In other words, using a repeated application of Parlett's method, a solution is given to the problem of completing the unitary k-Hessenberg matrix given its first k columns.