# Is every inner product defined by a matrix? Intuition?

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.

• Does your definition of "inner product" require positive definiteness (i.e. $\langle v,v \rangle > 0$ for all $v \ne 0$) ? – Ted Mar 31 '18 at 20:15
• The domain of your inner product is... – Martín-Blas Pérez Pinilla Mar 31 '18 at 20:20

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