I am struggling with understanding this:
For an inner product of $\mathbb{R}^3$ defined by $\langle x,y\rangle = 2x_1y_1 -x_1y_2 -x_2y_1 + 5x_2y_2$ the matrix relative to the standard basis is:-
$$\begin{pmatrix}2&-1\\-1& 5\end{pmatrix}$$
if the substitutions $$x_1 = (2/3)x_1' + (1/3)x_2'$$ $$x_2 = (1/3)x_1' - (1/3)x_2'$$
and $$y_1 = (2/3)y_1' + (1/3)y_2'$$ $$y_2 = (1/3)y_2' - (1/3)y_2'$$ are made,then the inner product takes the simple form $\langle x,y\rangle = x_1'y_1' + x_2'y_2' = x'^ty'$ . I understand why it works and I understand the use of eigenvectors to form an orthonormal vectors. Have tried this but the eigenvalues are messy.
How do I arrive at the above substitutions?
Define $\langle x,y\rangle_A:=x^tAy$ for a positive definit symmetric matrix $A$. Find a square root $N$ of $A$, i.e., $N^2=A$. (You know how to do this?). Then it is easy to derive that $$\langle x,y\rangle_A=\langle Nx,Ny\rangle_I,$$ where $I$ is the identity matrix.
In your example: $A=\begin{pmatrix}2&-1\\-1&5\end{pmatrix}$, $N=\begin{pmatrix}1&1\\ 1&-2\end{pmatrix}$ and btw. $N^{-1}=\frac 13 \begin{pmatrix}2&1\\ 1&-1\end{pmatrix}$.
Michael