Incoming links: Inner product
Given a linear operator over a space that has a inner product defined, we define the adjoint operator (the symbol is called "dagger") as the unique operator that satisfies:
The group of all transformations that preserve some bilinear form, notable examples:
- orthogonal group preserves the inner product
- unitary group preserves a Hermitian form
- Lorentz group preserves the Minkowski inner product
A metric is a function that give the distance, i.e. a real number, between any two elements of a space.
A metric may be induced from a norm as shown at: Section "Metric induced by a norm".
Because a norm can be induced by an inner product, and the inner product given by the matrix representation of a positive definite symmetric bilinear form, in simple cases metrics can also be represented by a matrix.
This form is not really an inner product in the common modern definition, because it is not positive definite, only a symmetric bilinear form.