1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Forming an orthogonal matrix whose 1st column is a given unit vector

  1. Nov 30, 2006 #1
    1. The problem statement, all variables and given/known data
    Show that if the vector [tex]\textbf{v}_1[/tex] is a unit vector (presumably in [tex]\Re^n[/tex]) then we can find an orthogonal matrix [tex]\textit{A}[/tex] that has as its first column the vector [tex]\textbf{v}_1[/tex].

    3. The attempt at a solution
    This seems to be trivially easy. Suppose we have a basis [tex]\beta[/tex] for [tex]\Re^n[/tex]. We may apply the Gram-Schmidt orthogonalization process to [tex]\beta[/tex] with [tex]\textbf{v}_1[/tex] as the generating vector and normalise the resultant orthogonal basis to obtain an orthonormal basis [tex]\gamma[/tex]. Choose [tex]\textit{A}[/tex] such that the elements of [tex]\gamma[/tex] comprise the columns of [tex]\textit{A}[/tex].



    So what am I overlooking?
    Last edited: Nov 30, 2006
  2. jcsd
  3. Dec 1, 2006 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    Nothing- although I would say "Let [itex]\beta[/itex] be a basis for R containing v1..."
  4. Dec 1, 2006 #3


    User Avatar
    Homework Helper

    Another candidate is any rotation matrix which rotates the vector [1,0,0,...] into v_1.
  5. Dec 1, 2006 #4
    Thanks. I thought I had made a huge assumption somewhere.

    StatusX, yes that's a simpler possibility indeed.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Forming an orthogonal matrix whose 1st column is a given unit vector