Chatbot math/Bard/24.02/Matrix versus tensor

How does wikipedia distinguish between a matrix and a rank 2 tensor?

This page documents a conversation between myself and Bard on on 23 January 2004. My comments are in code format. Let me begin with three summary comments:

Overall, I am impressed with Bard's answer. It was a coherent effort to look for differences in how tensors and matrices are discussed.
My biggest complaint is Bard's inability to see the "big picture". Bard should have began by saying that this is not a well-posed question. Precise statements are essential to mathematical discussion. Bard treated this question almost as if it were asked to explain the difference between a real and imaginary number. Bard inability to grasp this led to its first blunder by saying "...Wikipedia highlights the key distinction between them". There is no clear distinction between the terms.--Guy vandegrift (discuss • contribs) 23:54, 23 January 2024 (UTC)
This conversation is consistent with previous ones I have had with Bard. Bard reminds me of that new employee who acts confident in an effort to mask deep insecurity. It is not uncommon for people who enter the stage in this fashion to evolve into valuable contributors to the enterprise.

--13:41, 27 January 2024 (UTC)

The following question was put to Bard:

How does wikipedia distinguish between a matrix and a rank 2 tensor?

While matrices and rank 2 tensors share a similar rectangular representation, Wikipedia highlights key distinctions between them:

1. Tensors as Multilinear Maps vs. Matrices as Linear Maps:I do not consider this a valid distinction, and suspect that Bard just looked at how many time a matrix was multiplied by column vector and jumped to a conclusion.

Tensors: Represent multilinear maps, taking multiple vectors as input and producing a scalar. A rank 2 tensor takes two vectors as input.
Matrices: Represent linear maps, taking a single vector as input and producing a vector as output.

2. Coordinate Independence vs. Basis Dependence:This might be valid, but is a trivial distinction. In mathematics, something is defined in the language of mathematics, and therefore is "inherently tied" to the basis that defined it. In physics, the tensor is essential a thought experience, e.g., "if I subject this system to a force, what will be the displacement?"

Tensors: Defined intrinsically, independent of any basis. Their components in a specific basis form a matrix-like array, but this representation can change under a change of basis.
Matrices: Inherently tied to a specific basis. Their entries represent linear transformations within that basis.

3. Transformation Under Coordinate Changes:As with the previous comment, likely to be true.

Tensors: Transform according to specific rules under a change of coordinates, ensuring their intrinsic meaning remains consistent.
Matrices: Don't inherently have transformation rules under a change of basis. Their entries might need to be re-calculated for a new basis.

4. Rank and Order:This is apparently true. See this discussion on mathworld.wolfram.com

Rank of a Tensor: Refers to the number of indices required to represent its components (e.g., a rank 2 tensor has two indices).
Order of a Tensor: Often used interchangeably with rank, but can also refer to the tensor's "degree," which is the sum of the orders of its arguments.

5. Matrix Rank vs. Tensor Rank:Also appears to be true

Matrix Rank: Related to linear independence of rows or columns.
Tensor Rank: The minimum number of simple tensors (rank 1 tensors) that can be summed to create the tensor.

6. Notation:Definitely true, though Einstein summation notation is not the only common way to describe a tensor.

Tensors: Often employ index notation with Einstein summation convention (e.g., T_ij) for clarity and compactness, especially for higher-order tensors.
Matrices: Commonly use matrix notation with parentheses or brackets (e.g., A = [a_ij]).

In summary, while matrices can be viewed as a specific type of rank 2 tensor, tensors offer a more general and abstract framework for representing multilinear relationships in a coordinate-independent manner. I have to admit that while this answer does not impress me very much, I can't think of anything that is much better.Guy vandegrift (discuss • contribs) 00:31, 24 January 2024 (UTC)

Could you give me links to the articles?

I only glanced at these links. Students could be assigned the task of factchecking these claims

Certainly! Here are some resources from Wikipedia that further delve into the distinctions between matrices and rank 2 tensors:

Tensor (intrinsic definition): w:Tensor
- This article establishes the foundation for understanding tensors as multilinear maps and highlights their intrinsic nature, independent of any chosen basis.
Rank (linear algebra): https://mathworld.wolfram.com/TensorRank.html
- This article clarifies the concept of rank, both for matrices and tensors, and distinguishes between different interpretations like column/row rank and tensor rank.
Tensor product: w:Tensor_product
- This article discusses the tensor product, a crucial operation for constructing higher-order tensors from lower-order ones, illustrating how matrices as rank 2 tensors can be seen as a specific case.
Differences between a Matrix and a Tensor: w:Matrix_calculus
- This article summarizes the key differences in a table format, providing a quick overview of the distinguishing features.
Geometric Transformations: w:Transformation_geometry
- This article covers various transformations, including those relevant to tensors and coordinate changes. It showcases how tensors behave differently from matrices under such transformations.
Einstein summation convention: w:Einstein_notation
- This article explains the Einstein summation convention, commonly used in tensor notation, for representing implicit summations across repeated indices, enhancing tensor expressions.

I hope these resources provide a deeper understanding of the concepts and clarify the subtle differences between matrices and rank 2 tensors. Feel free to ask if you have any further questions!