Suppose we have a matrix $A$ over which an inner product is defined. Let us denote this inner product by $\langle, \rangle_A$.
Now let us suppose that this matrix $A$ is symmetric. Then there exists a matrix $Q$ such that $Q^TAQ$ is a diagonal matrix. For any two vectors $v$ and $w$, is $\langle v,w\rangle_A=\langle v,w\rangle_{Q^TAQ}$?
In other words, is inner product preserved when we change the basis of $R^n$?