Is a projection operator hermitian?

In summary, the conversation discusses Theorem 1.2 from Georgi's book on Lie Algebras, which proves that every finite group is completely reducible. It also mentions the use of projection operators and their properties, including being Hermitian. The speaker suggests reviewing the properties of projection operators for further understanding.
  • #1
krishna mohan
117
0
I was reading Lie Algebras in Physics by Georgi......second edition...

Theorem 1.2: He proves that every finite group is completely reducible.

He takes

[tex]PD(g)P=D(g)P[/tex]


..takes adjoint...and gets..

[tex]P{D(g)}{\dagger} P=P {D(g)}{\dagger} [/tex]

So..does this mean that the projection operator P is hermitian?
 
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  • #2
krishna mohan said:
I was reading Lie Algebras in Physics by Georgi......second edition...

Theorem 1.2: He proves that every finite group is completely reducible.

He takes

[tex]PD(g)P=D(g)P[/tex]


..takes adjoint...and gets..

[tex]P{D(g)}{\dagger} P=P {D(g)}{\dagger} [/tex]

So..does this mean that the projection operator P is hermitian?

I think he is assuming P is Hermitian (as it must be; if you want to think of this in terms of QM, it leaves all states unchanged, so the eigenvalue associated with the operator is '1', a real number - any operator which outputs a real eigenvalue is Hermitian)
 
  • #3
All projection operators are hermitian. You might want to review the properties of projection operators here.
 
  • #4
Thanks a lot!:smile:
 

Related to Is a projection operator hermitian?

1. What is a projection operator?

A projection operator is a mathematical concept used in linear algebra and functional analysis. It is a linear transformation that maps a vector onto a subspace, essentially "projecting" the vector onto that subspace.

2. What does it mean for a projection operator to be Hermitian?

A Hermitian projection operator is one where the adjoint of the operator is equal to the operator itself. In other words, the operator is self-adjoint, meaning it is symmetric with respect to the inner product of the vector space it operates on.

3. Why is it important for a projection operator to be Hermitian?

A Hermitian projection operator has several important properties that make it useful in mathematical and scientific applications. These include being idempotent (meaning applying the operator twice results in the same vector), having real eigenvalues, and being unitary (meaning its inverse is equal to its adjoint).

4. How can I determine if a projection operator is Hermitian?

To determine if a projection operator is Hermitian, you can use the following test: take the adjoint of the operator and multiply it by the original operator. If the result is equal to the original operator multiplied by its adjoint, then it is Hermitian. Additionally, you can check if all the eigenvalues of the operator are real, as this is another characteristic of Hermitian operators.

5. What are some examples of Hermitian projection operators?

Some common examples of Hermitian projection operators include the identity operator (which maps a vector onto itself), the orthogonal projection operator (which projects a vector onto a subspace perpendicular to a given vector), and the Fourier transform operator (which projects a function onto a set of orthogonal functions).

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