In this post, to prove that left multiplying a matrix $V$ by an orthogonal matrix $A$ preserves angles among columns of $V$, the author used the following equality:
$(Av_i)^T(Av_j)=v_i^TA^TAv_j = v_i^Tv_j$
Why use $(Av_i)^T(Av_j)$?
2026-04-02 22:07:05.1775167625
Preservation of angles
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Angles are measured by the dot product, and what that little computation shows is that the dot product of two vectors is unchanged after they have been transformed by the orthogonal matrix $A$.