Results 1 to 2 of 2

Math Help - Gram-Schmidt process to find last row of matrix

  1. #1
    Senior Member sfspitfire23's Avatar
    Joined
    Oct 2009
    Posts
    273

    Gram-Schmidt process to find last row of matrix

    Hi guys,

    Use the Gram-Schmidt process to find last row of the following orthogonal matrix where the first two rows are:

    [sqrt(3)/2, -1/sqrt(8), 1/sqrt(8)]

    and

    [0, 1/sqrt(3), sqrt(2)/sqrt(3)]

    (third row is [a,b,c]).

    Now, because the matrix is orthogonal, all of the rows are perpendicular and have length 1. To find the third row, call it q3, using Gram-Schmidt, we would do this-

    q3 = [a,b,c] - proj q1 ([a,b,c]) - proj q2 ([a,b,c])

    where q1 = [sqrt(3)/2, -1/sqrt(8), 1/sqrt(8)] and q2 = [0, 1/sqrt(3), sqrt(2)/sqrt(3)].

    When I try and brute force this thing and solve, I get crazy fractions etc. I must be missing something...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member MaxJasper's Avatar
    Joined
    Aug 2012
    From
    Canada
    Posts
    482
    Thanks
    54

    Lightbulb Re: Gram-Schmidt process to find last row of matrix

    p = \left(\begin{array}{ccc} \frac{\sqrt{3}}{2} & -\frac{1}{2 \sqrt{2}} & \frac{1}{2 \sqrt{2}} \\ 0 & \frac{1}{\sqrt{3}} & \sqrt{\frac{2}{3}} \\ a & b & c\end{array}\right)

    p=orthogonal which means:

    p^{\mathsf{T}}.p=I


    Orthogonalize each column, and solve above identity to find a,b,c making p:

    p = \left(\begin{array}{ccc} \frac{\sqrt{3}}{2} & -\frac{1}{2 \sqrt{2}} & \frac{1}{2 \sqrt{2}} \\ 0 & \frac{1}{\sqrt{3}} & \sqrt{\frac{2}{3}} \\ \frac{3}{2} \sqrt{\frac{1}{119} \left(3+8 \sqrt{2}\right)} & \frac{16 \sqrt{357 \left(3+8 \sqrt{2}\right)}-3 \sqrt{714 \left(3+8 \sqrt{2}\right)}}{1428} & \frac{-16 \sqrt{357 \left(3+8 \sqrt{2}\right)}+3 \sqrt{714 \left(3+8 \sqrt{2}\right)}}{1428}\end{array}\right)
    Last edited by MaxJasper; October 20th 2012 at 10:49 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Gram-Schmidt process Help
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: July 10th 2011, 03:06 PM
  2. Gram-Schmidt Process
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 4th 2009, 10:42 PM
  3. Gram Schmidt Process
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 28th 2009, 11:16 PM
  4. Gram-Schmidt Process
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 31st 2009, 02:15 PM
  5. Gram-Schmidt process
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 31st 2009, 03:45 PM

Search Tags


/mathhelpforum @mathhelpforum