A Hermitian operator is a mathematical operator that satisfies the property of self-adjointness, meaning that it is equal to its own conjugate transpose. In other words, if we take the complex conjugate of the operator and transpose it, we will get the same operator back.
A normal operator is an operator that commutes with its adjoint, while a Hermitian operator is a special case of a normal operator that also satisfies the property of self-adjointness. This means that a Hermitian operator will always commute with its own conjugate transpose, while a normal operator may or may not.
To determine if an operator is Hermitian, we need to check if it satisfies the property of self-adjointness. This can be done by taking the conjugate transpose of the operator and comparing it to the original operator. If they are equal, then the operator is Hermitian.
Hermitian operators are commonly used in quantum mechanics to represent physical observables such as energy, momentum, and spin. They are also used in linear algebra and functional analysis to study complex vector spaces and operators.
No, a non-square matrix cannot be a Hermitian operator. In order for an operator to be Hermitian, it must be square, meaning that it has the same number of rows and columns. This is because the conjugate transpose of a matrix can only be taken if the matrix is square.