🎳 Inner Product

A mapping is symmetric if and positive definite if for vector space , and . A bilinear mapping satisfies

and the same for the order of elements swapped.

A bilinear, symmetric, and positive definite mapping is called an inner product, with usually denoted as .

Note

One common inner product is the dot product,

Info

A less common definition of the inner product can be applied to functions and ,

Symmetric Positive Definite §

If 🍱 Matrix is symmetric positive definite, then

defines an inner product where and are coordinate representations with respect to basis , and .