Results 1 to 2 of 2

Thread: Orthonormal change of basis matrix

  1. #1
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782

    Orthonormal change of basis matrix

    Suppose I have a matrix A w.r.t some basis v_1,....,v_n.

    Further suppose that this matrix undergoes a basis change to \underline{v}_1,...., \underline{v}_n where each \underline{v}_i is orthonormal to another (ie. \underline{v}_1,...., \underline{v}_n is an orthonormal basis). Call this matrix B (which has columns \underline{v}_1,...., \underline{v}_n).

    Let the change of basis matrix be Q.

    Is it true that A=Q^T B Q?

    Is it also true that det(Q)=1?

    (P.S I've done a question and this is one of the facts I used. I'm wondering if it really is a fact!).

    Thanks in advance!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    The change of basis matrix is orthogonal, which implies that its determinant is \pm1. It will be +1 if the basis change is orientation-preserving, and 1 if it is orientation-reversing. The formula A = Q^{\textsc t}BQ is correct.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Change of coordinate matrix, new basis
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Jul 13th 2011, 09:39 PM
  2. Compute the change of basis matrix
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Mar 19th 2011, 11:49 AM
  3. change of basis matrix from B1 to B2.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Feb 13th 2011, 09:42 AM
  4. Change of basis and model matrix
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Nov 15th 2009, 09:01 AM
  5. Change of Basis Matrix
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: Jun 20th 2009, 01:19 PM

Search Tags


/mathhelpforum @mathhelpforum