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In one proof of some theorem regarding normal distribution there is used I think an idenitity which states

$$ (ABy,By)=(B^{-1}ABy,y)$$ where A is symetric and positive, B is orthogonal and y is a vector.

Can someone explain to me why is it so?

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I assume $(\cdot, \cdot) $ denotes the scalar product. An orthogonal matrix $B$ preserves the scalar product by definition, i.e. $$(Bx,By)=(x,y).$$ Since $B^{-1}$ is also orthogonal, then $$(ABy, By) = (B^{-1}ABy,B^{-1}By)=(B^{-1}ABy,y).$$

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It follows easily from
1. $(u,v)=u^tv$, where $u^t$ is the transpose of $u$,
2. $(CD)^t=D^tC^t$,
3. $C$ symmetric iff $C=C^t$, and
4. $C$ orthogonal iff $C^t=C^{-1}$.

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    $\begingroup$ The fact that $A$ is symmetric is not needed. $\endgroup$ – Joppy Oct 6 '18 at 9:58
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An equivalent definition of an orthogonal matrix is one for which $(Bv, Bw) = (v, w)$ for all vectors $v$ and $w$. You can check that if $B$ is orthogonal, then $B^{-1}$ exists and is orthogonal. Can you work it out from there?

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