A linear Operator is a map satisfying

In the above:

Operators are a generalisation of matrices: They act on Vector Spaces linearly, without being attached to any specific basis. When working with finite Vector Spaces, the choice of a basis allows us to represent the Operator as a matrix. This won’t be always possible: Operators in infinite dimensional spaces do not always have a Matrix representation

Relations

The diagram below aims to make relations between Operators and their Matrix representation memorable. An arrow means an implication.

  • Improve the relations below: They are lacking at the moment.
stateDiagram
	Hermitian: Hermitian
	Positive: Positive
	Bounded: Bounded Operators
	Real: Has real valued eigenvalues
	SelfAdjoint: Self Adjoint
	Spectral: Has a spectral decomposition
	Symmetric: Symmetric
	Unitary: Unitary
	Symmetric --> Hermitian : If Bounded
	Bounded --> Hermitian : If Symmetric
	Positive --> Spectral : If compact
	Hermitian --> Real
	SelfAdjoint --> Symmetric

See more details on:

Properties