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Math Help - Orthonormal change of basis matrix

  1. #1
    Super Member Showcase_22's Avatar
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    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!
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    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.
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