I am trying to solve a problem on the dot product, and I do some manipulations and come to the conclusion that
$\langle x, A^{T} Ax \rangle = \langle x,x \rangle$. $x$ is a column vector with $n$ rows, and $A$ an $n \times n$ matrix.
I know in general that if $a \cdot b = a \cdot c$, then this does not mean that $b = c$. However, from the above concerning the matrix $A$, can I conclude that the only way the left and right hand sides are equal is when $A^{T}A = I$, the identity matrix? Otherwise how can the two quantities be equal?
If the statement were not true, I would be glad if someone could provide a counter example.
Thanks