How does changing an inner product change the angle between two vectors? Very specifically, if $\langle x,y\rangle_A = \langle Ax,y\rangle < 0$ where $A\succ 0$, does that mean than $\langle x,y\rangle<0$? Basically, if $x$ and $y$ are at an obtuse angle in the $A$-inner product, are they guaranteed to be obtuse in the original inner product?
How does changing an inner product change the angle between two vectors?
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real-analysis
linear-algebra
vector-spaces
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1No, consider $x=(1,\epsilon)$, $y=(\epsilon,1)$, and $A=\begin{bmatrix}1&-1/2\\-1/2&1\end{bmatrix}$. – 2017-01-19
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0Thanks! That indeed does break it. Do you know of any property that could be imposed on $A$ where it would in fact be true? – 2017-01-19
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0No, for any s.p.d. $A$ that is not a multiple of $I$ there exist two vectors $x,y$ that form a counterexample. I'll post an proof when I have time, or you can try proving it yourself. – 2017-01-19