How to normalize vector with non-standard scalar product?

Question

Let's say I have $V=\mathbb{R}$ and the scalar product is $<X,Y> = X^t*A*Y$ and $A = \begin{bmatrix} 2 & 1 \\ 1 & 2\\ \end{bmatrix}$

Now I want to compute an orthonormal basis. Can I use the Gram-Schmidt process? And how would I start? Because If I just choose the first vector in the Gram-Schmidt process to be the standard $v = (1,0)^t$ this won't be normalized. To normalize a vector can I always divide it by the square root of its scalar product, no matter how the scalar product is defined?

Answer 1

At some step of the GS process, you will need to form linear combinations such as

$$X-\lambda U-\mu V$$ where $U,V$ are previously built orthogonal vectors, and make them orthogonal to $U$ and to $V$.

You write $$U^TA(X-\lambda U-\mu V)=U^TAX-\lambda U^TAU-\mu U^TAV=U^TAX-\lambda U^TAU=0$$ giving

$$\lambda=\frac{U^TAX}{U^TAU}.$$

And similarly

$$\mu=\frac{V^TAX}{V^TAV}.$$

You can indeed normalize by the square root of the scalar square.

Answer 2

Yes. The process is independent of what bilinear form you are using.

For example, starting with $[1,0]$ and $[0,1]$, your first vector would be $[\frac{1}{\sqrt{2}},0]$, and following the Gram-Schmidt process the second vector becomes $[\frac{-\sqrt{6}}{6},\frac{\sqrt{6}}{3}]$.