how can I prove matrix H is orthogonal? Don't know where should I start with.
$$H = I - 2\frac{vv^T}{v^Tv}$$
2026-03-29 11:08:21.1774782501
how to prove this matrix is orthogonal
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Hint to get you started:
Evaluate $$\left(I-2\frac{vv^T}{v^Tv}\right)^T\left(I-2\frac{vv^T}{v^Tv}\right)$$
Edit:
\begin{align}&I-4I\frac{vv^T}{v^Tv}+4\frac{vv^T}{v^Tv}\frac{vv^T}{v^Tv}\\ &=I-4\frac{vv^T}{v^Tv}+4\frac{v(v^Tv)v^T}{(v^Tv)^2}\\ &=I-4\frac{vv^T}{v^Tv}+4\frac{(v^Tv)vv^T}{(v^Tv)^2}\\ &=I-4\frac{vv^T}{v^Tv}+4\frac{vv^T}{v^Tv}\\&=I\end{align}