Ciro Santilli @cirosantilli 37

Given a linear operator $A$ over a space $S$ that has a inner product defined, we define the adjoint operator $A^{†}$ (the $†$ symbol is called "dagger") as the unique operator that satisfies:

The group of all transformations that preserve some bilinear form, notable examples:

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.

An inner product $x\cdot y$ induces a norm with: