Array Representation Of A General Tensor Question

In summary, eigenchris's "Tensors for Beginners" video series covers the following: 1) the array for Q, which is a row of rows of columns; 2) the different types of tensor products that can be represented this way; and 3) how to represent a tensor product this way.
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Vanilla Gorilla
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TL;DR Summary
My question is just if the points are correct and true statements; if not, how could I rewrite them? :)
Some people like to write tensor products as 3d arrays, but that inherently means we lose information when compared to the 2D representation
That’s because in the 2D representation, we see this is a (1,2)-tensor, because there's 1 column aspect and 2 row aspects.
A (m,n) tensor can be represented by m column aspects and n row aspects, when converted to array form.
So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I, in the position of a complete beginner, am taking notes on it, and I just wanted to make sure I wasn't misinterpreting anything.

At about 5:50, he states that "The array for Q is a row of rows of columns. And some people like to think that since there are three parts in this tensor, they think that we should visualize this tensor array instead as a 3d cube, like over here. But I don't like to do that. Because when we visualize it this way, we lose out on how many vector parts and how many covector parts there are. And we sort of lose information about what type of tensor This is. But when I write the tensor out like this as a row of rows of columns, I can still see by looking at this, that this is a (1,2)-tensor, because there's one column aspect and two row aspects."
Regarding this, I interpreted it to imply the points below.
My question is just if the points are correct and true statements; if not, how could I rewrite them? :)
  1. Some people like to write tensor products as 3d arrays, but that inherently means we lose information when compared to the 2D representation
  2. A (m,n) tensor can be represented by m column aspects and n row aspects, when converted to array form.
Any help is much appreciated!
P.S., I'm not always great at articulating my thoughts, so my apologies if this question isn't clear.
P.P.S., I know this isn't high school material, but I am currently in high school, which is why I made my level "Basic/high school level"
 
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  • #2
The points are correct. A (1,2) tensor, a (2,1) tensor, a (3,0) tensor and a (0,3) tensor are all different, but can all be mapped to a 3D block of numbers. Having just a 3D block of numbers doesn't tell you which of those four types of tensor it represents. So it has lost that information about the tensor.
 
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Likes Vanilla Gorilla
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Thank you! Your response is much appreciated! :)
 
H2: What is an array representation of a general tensor?

An array representation of a general tensor is a way of organizing and storing multidimensional data in a tabular format. It is commonly used in mathematics and computer science to represent data structures and perform operations on them.

H2: How is a tensor represented as an array?

A tensor can be represented as an array by using indices to denote the dimensions of the tensor. For example, a 2-dimensional tensor can be represented as a 2D array with rows and columns, while a 3-dimensional tensor can be represented as a 3D array with rows, columns, and depth.

H2: What is the difference between a tensor and an array?

A tensor is a mathematical object that represents a multidimensional array of numbers, while an array is a data structure used to store and organize data in a tabular format. Tensors can have any number of dimensions, while arrays are typically limited to two or three dimensions.

H2: How is a tensor represented in programming languages?

In programming languages, tensors are typically represented as multidimensional arrays or matrices. Some languages, such as Python and MATLAB, have built-in libraries for handling tensors and performing operations on them.

H2: What are some applications of array representation of tensors?

Array representation of tensors is used in various fields such as physics, engineering, machine learning, and image processing. It is especially useful for representing and manipulating large datasets and performing complex mathematical operations on them.

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