An example of non euclidean inner product

Question

Please give me an example of non euclidean inner product.Is there any method to construct such an inner product?

Answer 1

Choose a symmetric matrix $A$ with determinant zero. A non-eucliden inner product can now be defined between two vectors $u,v$ by matrix multiplication by $v^tAu$ (interpreting the resulting $1\times1$ matrix as a real number)

Answer 2

A Euclidean space is a pair $(V,g)$ where $V$ is a finite-dimensional real vector space and $g$ is an inner product, if by "inner product" we mean a bilinear map $g \colon V \times V \to \mathbf{R}$ which is symmetric and positive definite.

However, in some contexts one doesn't have positive definiteness, say special relativity with Minkowski space $V \cong \mathbf{R}^4$ and $$ g((t_1,x_1,y_1,z_1),(t_2,x_2,y_2,z_2)) = -t_1 t_2 + x_1 x_2 + y_1 y_2 + z_1 z_2 , $$ and some people still use the phrase "inner product" for this more general type of bilinear map. Someone who follows this convention might use the phrase "Euclidean inner product" to refer to the good old positive definite kind of inner product, and say "non-Euclidean" about something which is not positive definite.

If you consider complex vector spaces, you have a concept of inner product where the map isn't even bilinear, but linear in one factor and conjugate-linear in the other (but still positive definite). Perhaps the word "non-Euclidean" can be used in this context too, to distinguish it from the real case.

I haven't thought much about it, and I don't think it's worth making a big deal out of it, since it's usually clear from context what kind of inner product one is talking about anyway.