How to represent operator in matrix form

In summary, to represent an arbitrary operator O in a general matrix form that preserves its properties, you need a basis of Hilbert space and the matrix elements are given by O_{jk}=\langle j|\hat{O} k \rangle. This representation will result in an Hermitean matrix if the operator is selfadjoined.
  • #1
phyin
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I'm given some arbitrary operator call it O, how do I represent it in general matrix form while it still preserves the properties of the operator.

ex. if operator is hermitian how to i represent a most general matrix representation so it preserves properties of a hermitian matrix.
 
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  • #2
You need a basis of Hilbert space, [itex]\{|n \rangle \}_{n \in \mathbb{N}}[/itex], e.g., the harmonic-oscillator-energy eigen states. Then the matrix elements of an arbitrary operator, [itex]\hat{O}[/itex] are given by

[tex]O_{jk}=\langle j|\hat{O} k \rangle.[/tex]

It's easy to verify that this is an Hermitean matrix, if [itex]\hat{O}[/itex], is selfadjoined.
 

1. What is an operator in matrix form?

An operator in matrix form is a representation of a mathematical operation using matrices. It is a useful tool in linear algebra for performing calculations and solving equations.

2. How is an operator represented in matrix form?

To represent an operator in matrix form, the operator is written as a matrix with the same dimensions as the vector or matrix that it is operating on. The elements of the matrix correspond to the coefficients of the operation.

3. What are the advantages of representing an operator in matrix form?

Representing an operator in matrix form allows for easier manipulation and calculation of the operation. It also allows for the use of matrix algebra, which can be more efficient and intuitive than traditional algebraic methods.

4. Can all operations be represented in matrix form?

No, not all operations can be represented in matrix form. Only linear operations can be represented in matrix form, where the output is a linear combination of the input. Non-linear operations, such as exponentiation or logarithms, cannot be represented in matrix form.

5. How is the inverse of an operator represented in matrix form?

The inverse of an operator in matrix form is represented by taking the inverse of the matrix that represents the operator. This can be found using various methods, such as Gaussian elimination or the adjugate matrix method.

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