In a matrix representation, you can write the original equation as a sum of products using matrix multiplication rules. Take the complex conjugate, and replace the conjugates of the elements of A with elements of B. Then from the relationship between elements of <g| and |g>, <f| and |f> you have all you need.rayveldkamp said:Hi,
I am trying to show that |g> = A|f> implies
<g| = <f|B
where A is an operator and B is its Hermitian conjugate.
I think my problem is with notation, but i have not been able to show this as yet.
Thanks
Ray
rayveldkamp said:Hi,
I am trying to show that |g> = A|f> implies
<g| = <f|B
where A is an operator and B is its Hermitian conjugate.
I think my problem is with notation, but i have not been able to show this as yet.
Thanks
Ray
As far as I'm aware, that is how the hermitian conjugate (or adjoint) is defined - though the dual correspondence for A|a> !HallsofIvy said:That's pretty much the definition of Hermitian conjugate, isn't it?
This statement is a mathematical expression that refers to the proof that the inner product of |g> and |f> is equal to the inner product of A|f> and B|g>. In other words, it shows that the inner product of two vectors is preserved under linear transformations.
This proof is relevant in many areas of science, including physics, engineering, and computer science. It is commonly used to prove theorems and solve problems in linear algebra and quantum mechanics.
Sure! For example, in quantum mechanics, this proof is used to show that the expectation value of an observable operator is independent of the choice of basis. This is a fundamental concept in quantum mechanics and is crucial in understanding the behavior of particles at the microscopic level.
The key components of this proof are the properties of inner products, specifically the bilinearity and conjugate symmetry properties. These properties allow us to manipulate the inner product of vectors and show that it is preserved under linear transformations.
This proof is considered a basic concept in mathematics, specifically in the field of linear algebra. It is often taught in introductory courses and is essential for understanding more advanced concepts in linear algebra and quantum mechanics.