Representation of Inner Products ( Inner Product Matrix).

Representation of Inner Products ( Inner Product Matrix).

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?