# Matrix representing projection operators

• SouthQuantum
In summary, the conversation discusses the matrix representation of a projection operator and whether the wavefunction needs to be normalized before determining the projection operator. It is stated that for Ψ1, which is normalized, a 3x3 Hermitian matrix can be calculated by multiplying row and column matrices. However, there is a question about whether Ψ2, which is not normalized, needs to be normalized before the same process can be done. It is clarified that technically, the wavefunction should be normalized to build a projection operator, but it would still be Hermitian either way. The conversation concludes with thanks for the help and confirmation that the wavefunction should be normalized.
SouthQuantum
Hey everyone! I have a question regarding the matrix representation of a projection operator. Specifically, does the wavefunction have to be normalized before determining the projection operator? For example:

|Ψ1> = 1/3|u1> + i/3|u2> + 1/3|u3>

|Ψ2> = 1/3|u1> + i/3|u3>

Ψ1 is obviously normalized and Ψ2 isnt

Now to calculate the matrix that represents the projection operator, just make row and column matrices and multiply out. For Ψ1:

| 1/3|u1> |
| i/3|u2> | * (| 1/3|u1> i/3|u2> 1/3|u3> |)
| 1/3|u3> |

Which gives a 3x3 Hermitian matrix.

My question is do I have to normalize Ψ2 in order to do the same process? The result is a Hermitian operator either way, I believe, but I just want to know if 'technically' it needs to be normalized first.

thanks for any input!

Timmy

ps: I realize this is a little 'homeworky' and if it is too much so, I apologize for posting it in the wrong place.

yes if you want to build a projection operator out of a wavefunction then the wavefunction has to be normalized, but it's also true that it's Hermitian either way. A projection operator P should obey PP=P, so |$$\psi$$><$$\psi$$|$$\psi$$><$$\psi$$|=|$$\psi$$><$$\psi$$| requires <$$\psi$$|$$\psi$$>=1

Alright! Thanks for the help--I appreciate it!

Timmy

## 1. What is a projection operator?

A projection operator is a mathematical object that maps a vector space onto a subspace, by "projecting" the vectors onto the subspace. It can also be thought of as a transformation that projects a vector onto a lower-dimensional subspace.

## 2. How is a matrix used to represent a projection operator?

A projection operator can be represented by a square matrix with the same number of rows and columns as the dimensionality of the vector space. The entries of the matrix are determined by the projection of the basis vectors onto the subspace.

## 3. What is the significance of the matrix elements in a projection operator?

The matrix elements in a projection operator represent the inner products between the basis vectors and their projections onto the subspace. These elements are used to determine the projection of any vector onto the subspace.

## 4. Can a projection operator have more than one matrix representation?

Yes, a projection operator can have multiple matrix representations, depending on the choice of basis vectors for the vector space and the subspace. However, all these representations will have the same properties and will result in the same projections.

## 5. What are the properties of a projection operator?

A projection operator is idempotent, meaning that when applied multiple times, it results in the same output. It is also self-adjoint, meaning that its matrix representation is equal to its conjugate transpose. Additionally, the eigenvalues of a projection operator are either 0 or 1.

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