Prove that If A,B are 3x3 tensors, then the matrix C=AB is also a tensor

In summary, the question is asking for a proof that if two matrices have the same rank, then their product will also be of the same rank.
  • #1
ReuvenD10
9
1
Homework Statement
Prove that If A,B are 3x3 tensors, then the matrix C=AB is also tensor
Relevant Equations
the equations below in my solution
I try to solve but i have 1 step in the solution that I don't understand who to solve.

Below in the attach files you can see my solution, the step that I didn't make to prove Marked with a question mark.

thanks for your helps (:
 

Attachments

  • Doc Mar 29 2021.pdf
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  • #2
It would help a lot if you would type in your question here (see how you can type formulas on PF: https://www.physicsforums.com/help/latexhelp/) instead of forcing people to download your pdf. An explanation what a ##3\times 3## tensor should be, if not a ##3 \times 3## matrix would be helpful, too.

To me it reads as:
Show that the product of two square matrices of equal size is again a square matrix of the same size.
 
  • #3
fresh_42 said:
To me it reads as:
Show that the product of two square matrices of equal size is again a square matrix of the same size.

It's funny, I interpreted it slightly differently. That ##\mathcal{A}, \mathcal{B}, \mathcal{C}## are some rank-2 and 3-dimensional tensors and we are asked to prove the tensor transformation properties, i.e. to show that if an equation in matrix representation ##[\mathcal{C}]_{\beta_1} = [\mathcal{A}]_{\beta_1} [\mathcal{B}]_{\beta_1} ## holds with respect to basis ##\beta_1## then ##[\mathcal{C}]_{\beta_2} = [\mathcal{A}]_{\beta_2} [\mathcal{B}]_{\beta_2} ## holds with respect to basis ##\beta_2##. So for instance you have$$
\begin{align*}
\bar{c}_{\mu \nu} &= {\bar{a}_{\mu}}^{\gamma} \bar{b}_{\gamma \nu} = ({T^{\rho}}_{\mu} {T_{\sigma}}^{\gamma} {a_{\rho}}^{\sigma})({T^{\alpha}}_{\gamma}{T^{\beta}}_{\nu} b_{\alpha \beta}) \\

&= {T^{\rho}}_{\mu} {T^{\beta}}_{\nu} {a_{\rho}}^{\alpha} b_{\alpha \beta} \\

&= {T^{\rho}}_{\mu} {T^{\beta}}_{\nu} c_{\rho \beta}

\end{align*}
$$where the ##{T^i}_j## are the transformation coefficients from ##\beta_1## to ##\beta_2##.
 
  • #4
What is a matrix representation of a tensor? That doesn't make sense. A matrix is already a tensor. And any tensor other than ##\sum u_k\otimes v_k## isn't a matrix. ##3## by ##3## makes only sense for matrices. The rank should be completely irrelevant here.
 
  • #5
Sure, yes I'm still doing some mental gymnastics to try and understand what is required. Generally a tensor doesn't have a 'matrix representation', but you can naturally map rank-2 tensors in ##n##-dimensional space to ##n \times n## matrices, for ease of computation.

Anyway that's just how I interpreted it, I guess we need to wait for OP to explain what the question actually is asking.
 
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FAQ: Prove that If A,B are 3x3 tensors, then the matrix C=AB is also a tensor

1. What is a tensor?

A tensor is a mathematical object that represents a physical quantity with multiple components. It is often used to describe the relationship between different vectors and matrices in a multidimensional space.

2. How is a tensor different from a matrix?

A tensor is a more general mathematical object than a matrix. While a matrix has two dimensions (rows and columns), a tensor can have any number of dimensions. Additionally, the components of a tensor can be vectors, matrices, or other tensors, while the components of a matrix are always numbers.

3. How do you prove that the matrix C=AB is a tensor?

To prove that the matrix C=AB is a tensor, we need to show that it satisfies the properties of a tensor. This includes showing that it transforms according to the tensor transformation law, which states that the components of a tensor change in a specific way when the coordinate system is changed. We also need to show that the components of C are linear combinations of the components of A and B, and that it follows the tensor product rule.

4. What is the significance of proving that C=AB is a tensor?

Proving that C=AB is a tensor is important because it allows us to use the properties and operations of tensors to manipulate and analyze the matrix. This can be useful in various fields of science, such as physics, engineering, and computer science, where tensors are commonly used to model and solve complex problems.

5. Are there any real-world applications of tensors?

Yes, tensors have many real-world applications. They are used in physics to describe the relationship between physical quantities, in engineering to model stress and strain in materials, and in computer science for tasks such as image and speech recognition. Tensors are also used in machine learning and artificial intelligence algorithms for data analysis and pattern recognition.

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