Math Beyond Matrices: Is There Anything Else?

[danago]

I was just thinking, and thought this would be the best place to ask. So far in maths I've come across real numbers, then vectors in multiple dimensions, and then matrices. Is there anything beyond this? Is there anything in maths that goes to the next step and perhaps takes a 'cube' or numbers, in a similar way to how a matrix is a square array (or rectangular) of numbers? I guess it would be similar to a "3-D matrix". Sorry if i haven't really explained myself well.

Thanks,
Dan.

 

[danago]

Is this effectively what a tensor is? I have never looked at tensors before, but i remember somebody saying a tensor is a generalization of matrices/vectors etc.

 

[granpa]

I think it goes scalar-vector-tensor. beyond that I don't know.

maybe that's what they call a tensor of rank 2.

 

[granpa]

http://www.physlink.com/Education/AskExperts/ae459.cfm

actually a rank 2 tensor is just a regular tensor and a tensor of rank 1 is a vector. so te answer to your question is a tensor of rank 3.

 

[belliott4488]

Is this effectively what a tensor is? I have never looked at tensors before, but i remember somebody saying a tensor is a generalization of matrices/vectors etc.

Yes, that is the best answer to what you're describing, although it's not quite right to think of real numbers as a starting point.

A scalar (i.e. a one-dimensional number in the normal sense) is called a "rank 0 tensor"; a vector in n dimensions, which can be represented by a 1xn matrix, is a "rank 1" tensor; then you have quantities of higher rank, such as the inertia tensor representing the distribution of mass in a body, that can be represented by nxn matrices and are "rank 2" or higher tensors.

The important thing about all these tensors is that they are not simply arrays of numbers. What really defines them is the way they behave under rotations of the underlying coordinate systems. Not every 1xn array of numbers can be interpreted as a vector, and similarly not every nxn matrix represents a rank 2 tensor.

As for extensions beyond real numbers, have you studied complex numbers? One way to understand them is as vectors in the "complex plane", reflecting the fact that any complex number is an inherently 2-component object. They can also be represented as ordered pairs of real numbers with special rules for how to add, subtract, multiply, etc. them.

Complex numbers can then be extended to quaternions, which can be represented as ordered pairs of complex numbers, and as such have four component parts. Next is octonions, which have eight components, but that's as far as you can go. Each higher level of number loses some of the properties that make numbers useful (quaternions don't obey the commutative law of multiplication, and octonions don't obey the associative law of multiplication over addition), and going any higher become impossible (as far as I know, at least).

 

[danago]

Thanks for the replies everyone, answered my question spot on

 

[HallsofIvy]

Note, however, that "matrix" and "tensor" are very different things. I second order tensor corresponds more closely to a linear transformation from a two dimensional vector space to itself. With a given basis or coordinate system we can represent a such a linear transformation as a matrix. Vectors, linear transformations, and tensors are independent of the particular basis or coordinate system used. n-tuples of numbers, matrices, etc. are ways of representing those vectors, linear transformations, and tensors in a particular coordinate system.