Types of operators
There are certain types of linear operators that are special and need to be defined so that we can refer to them later in the book. You will also hear of them all the time in quantum computing.
Normal operators
Normal operators are ones that commute with their adjoint. For an operator Â, if:
(6)
then  is normal. They are important because a normal operator is diagonalizable, which is something we will consider later in the book. The following operators (Hermitian, unitary, positive, and positive semi-definite) are all normal operators.
A normal matrix represents a normal operator, and it commutes with its conjugate transpose. Let's look at an example normal matrix A:
Its conjugate transpose is:
Now, let's see if A commutes with its conjugate transpose...