Adjoint linear operator

In summary, the concept of an adjoint linear operator is defined as A^\dagger, where \langle x,Ay\rangle=\langle A^\dagger x,y\rangle. Physicists and mathematicians use different notations for the adjoint, but both refer to the same concept. A linear operator is self-adjoint if A^\dagger=A. To find the adjoint operator, one must use the definition and solve for A^\dagger.
  • #1
If L is the following first-order linear differential operator
L = p(x) d/dx
then determine the adjoint operator L* such that
integral from (a to b) [uL*(v) − vL(u)] dx = B(x) |from b to a|
What is B(x)?

sorry.. on my book there's only self-adjointness

i don't quiet understand what is the adjoint liear operator.

may someone solve this problem and tell me what exactly adjoint linear operator is ?
 
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  • #2
The adjoint [itex]A^\dagger[/itex] of a linear operator [itex]A[/itex] is defined by [itex]\langle x,Ay\rangle=\langle A^\dagger x,y\rangle[/itex]. Physicists write the adjoint as [itex]A^\dagger[/itex], mathematicians write it as [itex]A^*[/itex]. A is self-adjoint if [itex]A^\dagger=A[/itex].
 
  • #3
Fredrik said:
The adjoint [itex]A^\dagger[/itex] of a linear operator [itex]A[/itex] is defined by [itex]\langle x,Ay\rangle=\langle A^\dagger x,y\rangle[/itex]. Physicists write the adjoint as [itex]A^\dagger[/itex], mathematicians write it as [itex]A^*[/itex]. A is self-adjoint if [itex]A^\dagger=A[/itex].

how to find the A* then.. sry :(
 

1. What is an adjoint linear operator?

An adjoint linear operator is a type of linear transformation that maps a vector space to its dual space. It is also known as the adjoint of a linear operator, and is denoted by the symbol *.

2. How is an adjoint linear operator different from a regular linear operator?

An adjoint linear operator is different from a regular linear operator in that it maps a vector space to its dual space, while a regular linear operator maps a vector space to another vector space. Additionally, the adjoint operator preserves the inner product and orthogonality of vectors.

3. What is the importance of the adjoint linear operator in mathematics and science?

The adjoint linear operator is important in mathematics and science because it allows for the study of dual spaces and the preservation of inner products. It also has applications in fields such as physics, engineering, and computer science.

4. How is the adjoint linear operator related to the transpose of a matrix?

The adjoint linear operator is closely related to the transpose of a matrix. In fact, the transpose of a matrix can be seen as a special case of the adjoint operator, where the vector spaces are finite dimensional and the inner product is the standard dot product.

5. Can an adjoint linear operator exist for any type of vector space?

Yes, an adjoint linear operator can exist for any type of vector space, as long as the vector space has a defined inner product. This includes finite-dimensional and infinite-dimensional vector spaces, as well as real and complex vector spaces.

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