Exploring the Possibility of Cube Matrices in Linear Algebra

[Vorde]

I just had my last Linear Algebra class, and I didn't get a chance to ask the one question that has been bugging me ever since we started in earnest with matrices.

Why aren't there cube matrices? I mean, mathematical entities where numbers are 'laid out' in 3d not in 2d (not quite mathematically rigorous, but you get the idea). Obviously one could do this with successive matrices, but I wonder if more is to be gained by studying this object as a whole.

Is this a thing? Is there an obvious reason I'm missing as to why there is nothing to gain by doing this?

 

[pbuk]

There are, and they certainly are useful; physics wouldn't have got much beyond Newton without them. Google "tensors".

 

[Vorde]

Huh, you know I've known about tensors for a while but in a purely pop-science way (the only actual tensors I've been exposed to were in a five-minute digression by my teacher briefly explaining them), I hadn't ever been told to think of them that way!

 

[pbuk]

That's a shame, although you have actually been exposed to tensors for some time. Ordinary numbers (scalars), vectors and matrices are just special cases of tensors with 0, 1 and 2 dimensions respectively.

 

[blue_raver22]

I find this question very intriguing. While cube matrices may not be a commonly studied topic in linear algebra, there is definitely potential for exploring their properties and applications. In fact, there are already some areas of mathematics where cube matrices are used, such as in the study of quaternions and in computer graphics.

The reason why cube matrices may not be as widely discussed in linear algebra is because they can often be represented as a combination of 2D matrices, as you mentioned. However, studying them as a whole entity can provide insights and nuances that may not be apparent when looking at them separately.

Additionally, cube matrices can be useful in representing and solving problems in 3D space, which is relevant in many scientific fields such as physics, engineering, and computer science. By incorporating cube matrices into linear algebra, we can expand our understanding and application of this fundamental mathematical concept.

I encourage you to continue exploring the possibility of cube matrices in linear algebra and to share your findings with others in the scientific community. Who knows, your research could potentially pave the way for new developments and advancements in the field.