Stage 1 built the geometry an inner product supplies. This short topic opens Stage 2 by building the operator taxonomy the spectral theorem is stated for: the adjoint of a linear map, and the special classes of operator — self-adjoint, normal, unitary/orthogonal — defined by how they relate to their own adjoint. This is deliberately the shortest topic in the unit; its entire job is definitional groundwork for Topic 4, where the payoff (every self-adjoint operator has an orthonormal eigenbasis) is finally proved.
By the end of this topic you should be able to:
- construct the adjoint of a linear map from its matrix (conjugate transpose);
- verify whether an operator is self-adjoint, normal, or neither;
- verify whether an operator is unitary/orthogonal;
- explain why the three classes are nested (self-adjoint and unitary are both normal, but not each other).
In this lesson
- Contents
- The big picture
The full lesson is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Linear Algebra, Optimisation and Numerical Methods syllabus.