Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8897731 | Linear Algebra and its Applications | 2018 | 6 Pages |
Abstract
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 Iâqqâ in the special form Iâqqâ=LD2Lâ, where L is nÃ(nâ1) and lower triangular with 1's on its main diagonal and D2=diag(μ12,â¦,μnâ12) is positive definite, Parlett observed that for i>j the entries of LË=LD can be written as lËij=âqiqâ¾jμj/Ïj, where qâ¾j denotes the complex conjugate of qj and Ïi=âj=i+1n|qj|2, Ïn=0, and μi=Ïi/Ïiâ1, for i=1,â¦,n. Furthermore, one solution to this problem is U=[qLË]. 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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Michael Mackenzie,