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Thread: 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 $\displaystyle A$ w.r.t some basis $\displaystyle v_1,....,v_n$.

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

    Let the change of basis matrix be $\displaystyle Q$.

    Is it true that $\displaystyle A=Q^T B Q$?

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

    (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 $\displaystyle \pm1$. It will be +1 if the basis change is orientation-preserving, and 1 if it is orientation-reversing. The formula $\displaystyle A = Q^{\textsc t}BQ$ is correct.
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