Is every inner product defined by a matrix? Intuition?

Question

Given an inner product $\langle\cdot,\cdot\rangle\rightarrow \mathbb{R}$, can we say that all such inner product are defined by a positive-definite matrix?

Is there an intuitive explanation (e.g. change of basis, a projection, measure of similarity) that can help explain the inner product and why certain matrices can define it? Thanks.

Answer 1

You are supposing that the inner product is defined in $\Bbb R^n$. As an inner product is bilinear, it will exists a matrix $A$ s.t.

$$\langle x,y\rangle = x^T A y.$$ (See https://en.wikipedia.org/wiki/Bilinear_form#Coordinate_representation)

If you are supposing that your inner product is positive definite, the matrix $A$ will be positive definite by definition.

Answer 2

It all depends on the nature of your vector space and the inner product defined on that vector space.

For example, you may define an inner product on the vector space of real continuous functions by $$ <f,g >= \int_a^b f(x)g(x) dx$$

which is not represented by matrix multiplications.