# 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?

• I have edited it to incorporate LaTeX.But I am not sure about the last equation.Please verify if that's what you wanted to say. Sep 28 '13 at 10:46
• Yes. When the subs are made and the inner product is multiplied out according to the definition, then the resulting matrix of the inner product is the identity matrix. So there are no x1y2 and x2y1 terms. So the inner product is in a simpler form. My question is how do I arrive at the correct substitutions in a systematic way. Sep 28 '13 at 10:56

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. $N^{-1}=\frac 13 \begin{pmatrix}2&1\\ 1&-1\end{pmatrix}$.
• If $N^2=A$, then $x^tAy=x^tN^2y=(x^tN)(Ny)=(N^tx)^t(Ny)=(Nx)^t(Ny)$ since $N$ is symmetric. So define $x'=Nx$ and $y'=Ny$ to obtain $x^tAy=x'y'$. So my solution stays correct. Sep 30 '13 at 11:19