What is the equation for representing a linear operator in terms of a matrix?

In summary, the conversation discusses the representation of linear operators by matrices and the equation for A(e_i). The speaker uses the Einstein summation convention to explain how to expand y=Ax in basis vectors and defines A_{ij}=(Au_j)_i. The equation for A(e_i) should be Ae_i = \sum_{j}(Ae_{i})_j e_{j} = \sum_{j}A_{ji}e_{j}.
  • #1
Favicon
14
0
I'm working through a proof that every linear operator, [tex]A[/tex], can be represented by a matrix, [tex]A_{ij}[/tex]. So far I've got

[tex]Let \textbf{p}=\sum_{i}p_{i}\widehat{\textbf{e}}_{i}[/tex]
[tex]A(\textbf{p}) = \sum_{i}p_{i}A(\textbf{e}_{i})[/tex]

which is fine. Then it says that [tex]A(\textbf{e}_{i})[/tex] is a vector, given by:

[tex]A(e_{i}) = \sum_{j}A_{j}(p_{i})e_{j} = \sum_{j}A_{ji}e_{j}[/tex].

The fact that its a vector is fine with me, but I can't get my head around the equation for it. why does the operator acting on one of the base vectors depend on [tex]p_{i}[/tex]? Surely the base vectors are independent of [tex]p_{i}[/tex] and so should be any operation acting on them.
 
Physics news on Phys.org
  • #2
Indeed, they don't.
I would write it like
[tex]A(\hat e_i) = \sum_j (A_i)_j \hat e_j[/tex]
where Ai is some vector of coefficients.
 
  • #3
This is how I do this thing: Suppose [itex]A:U\rightarrow V[/itex] is linear, and that [itex]\{u_j\}[/itex] is a basis for U, and [itex]\{v_i\}[/itex] is a basis for V. Consider the equation y=Ax, and expand in basis vectors.

[tex]y=y_i v_i[/tex]

[tex]Ax=A(x_j u_j)=x_j Au_j= x_j (Au_j)_i v_i[/tex]

I'm using the Einstein summation convention: Since we're always supposed to do a sum over the indices that appear exactly twice, we can remember that without writing any summation sigmas (and since the operator is linear, it wouldn't matter if we put the summation sigma to the left or right of the operator). Now define [itex]A_{ij}=(Au_j)_i[/itex]. The above implies that

[tex]y_i=x_j(Au_j)_i=A_{ij}x_j[/tex]

Note that this can be interpreted as a matrix equation in component form. [itex]y_i[/itex] is the ith component of y in the basis [itex]\{v_i\}[/itex]. [itex]x_j[/itex] is the jth component of x in the basis [itex]\{u_j\}[/itex]. [itex]A_{ij}[/itex] is row i, column j, of the matrix of A in the pair of bases [itex]\{u_j\}[/itex], [itex]\{v_i\}[/itex].

Favicon said:
[tex]A(e_{i}) = \sum_{j}A_{j}(p_{i})e_{j} = \sum_{j}A_{ji}e_{j}[/tex]
This one should be

[tex]Ae_{i} = \sum_{j}(Ae_{i})_j e_{j} = \sum_{j}A_{ji}e_{j}[/tex]

Note that the first step is just to express the vector [itex]Ae_i[/itex] as a linear combination of basis vectors, and that [itex](Ae_i)_j[/itex] is just what I call the jth component.
 
Last edited:

1. What is a linear operator?

A linear operator is a mathematical function that maps one vector space to another vector space while preserving the linear structure of the original space. In simpler terms, it is a mathematical operation that takes in a vector and outputs another vector.

2. What is a matrix?

A matrix is a rectangular array of numbers or variables. It is used to represent linear transformations and solve systems of linear equations. It is an important tool in linear algebra and has applications in various fields such as engineering, physics, and economics.

3. How are linear operators and matrices related?

Linear operators and matrices are closely related as linear operators can be represented by matrices and vice versa. The matrix representation of a linear operator depends on the choice of basis for the vector space.

4. What is the difference between a linear operator and a nonlinear operator?

A linear operator follows the properties of linearity, such as the preservation of scalar multiplication and vector addition. On the other hand, a nonlinear operator does not follow these properties and can exhibit more complex behavior.

5. How are linear operators and matrices used in real-world applications?

Linear operators and matrices have various applications in real-world problems, such as image processing, data compression, and optimization. They are also used in mathematical modeling to describe and analyze physical systems and phenomena.

Similar threads

  • Linear and Abstract Algebra
Replies
3
Views
964
  • Linear and Abstract Algebra
Replies
9
Views
858
Replies
7
Views
2K
  • Linear and Abstract Algebra
Replies
7
Views
1K
Replies
27
Views
1K
  • Quantum Physics
Replies
3
Views
746
Replies
4
Views
1K
  • Linear and Abstract Algebra
Replies
6
Views
2K
  • Linear and Abstract Algebra
Replies
12
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
1K
Back
Top