Example of Kronecker Delta Identity in 3D Matrix R

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In summary, the conversation discussed the Kronecker Delta expression and how it can be used to represent a rotation matrix in a 3-dimensional space. The concept of summing over repeated indices was explained and an example using a rotation matrix was provided. The importance of understanding each step and working through the manipulations was emphasized.
  • #1
intervoxel
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Kronecker Delta expression

Please, give me an example of this identity using a 3 dimensional matrix R (maybe representing a rotation). My difficulty lies in the indices manipulation.

[tex]
R_{ii'}R{jj'}\delta_{i'j'} = \delta_{ij}
[/tex]

I know it is obvious, but I'm really stuck in my self-teaching. Thank you.
 
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  • #2
Are you summing over repeated indices, e.g.
[tex]R_{ii'} R_{jj'} \delta_{i'j'} \text{ means } \sum_{i', j'} R_{ii'} R_{jj'} \delta_{i'j'}?[/tex]
By the definition of matrix multiplication:
[tex]R_{ij} R_{kl} \delta_{jl}= R_{ij} R_{kj} = R_{ij} R_{jk}^T = R_{ij} R_{jk}^T = (R R^T)_{ik}[/tex]
This is important, you should convince yourself of each step. If you must, write it out in matrices and compare each step to the index notation, seeing how operations like multiplying two matrices or taking a transpose translate into operations on the indices.

If this is equal to [itex]\delta_{ik}[/itex] that means that [itex]R R^T[/itex] is the identity matrix. An example of such a matrix is, such as
[tex]R = \begin{pmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}[/tex]

If you want you can explicitly write out all the 9 components of the left hand side (let i and j run and sum over k), and show that you are really just multiplying R by R^T. Or if you are not sadistic, take a 2x2 matrix :)

Sorry if I am too general, but I think it is best if you work through the manipulations yourself.
 
  • #3
Thank you for the answer. That's exactly what I expected. :)
 

Related to Example of Kronecker Delta Identity in 3D Matrix R

What is the Kronecker Delta Identity?

The Kronecker Delta Identity is a mathematical concept used in linear algebra to represent the identity matrix. It is often denoted by the symbol δij and has a value of 1 when i = j and a value of 0 when i ≠ j.

What is a 3D Matrix?

A 3D matrix, also known as a three-dimensional matrix, is a collection of data organized in three dimensions. It is represented by a set of numbers arranged in rows, columns, and layers, and is commonly used in mathematics, computer graphics, and scientific computing.

How is the Kronecker Delta Identity used in a 3D Matrix?

In a 3D matrix, the Kronecker Delta Identity is used to create a diagonal matrix with ones along the main diagonal and zeros everywhere else. This is useful for performing calculations and transformations on the matrix.

Can you provide an example of the Kronecker Delta Identity in a 3D Matrix?

One example of the Kronecker Delta Identity in a 3D Matrix is a 3x3x3 matrix with the Kronecker Delta function along the main diagonal. It would look like this:

δ11 0 0

0 δ22 0

0 0 δ33

What are the applications of the Kronecker Delta Identity in 3D Matrix R?

The Kronecker Delta Identity in 3D Matrix R has many applications in mathematics, physics, and engineering. It is commonly used in vector and matrix operations, solving linear equations, and representing transformations in 3D space. It is also used in computer graphics, image processing, and machine learning algorithms.

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