# Generalization of the matrix concept

It has been some time since I left university...

In a not too formal language, an $n$-dimensional vector is an indexed set of numbers $\{i_1, ..., i_n\}$. A $n\times m$ matrix is a set of numbers with a two-dimensional index $\{i_{11},...,i_{n1},i_{m1},...,i_{mn}\}$.

What is the generalization of this, i.e. a set of numbers with an $n$-dimensional index? What is it called?

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en.wikipedia.org/wiki/Tensor – Tom Cooney Jun 8 '12 at 11:35
Yes! That's it. If you'd given this as an answer, I'd chosen it. Thanks a lot! – Axel Jun 10 '12 at 19:30

Actually, you should think of a vector as a function $$x: \{1, 2, 3, .... , n\}\rightarrow\mathbb{R}.$$ The right abstraction is to think of a vector or a matrix as a function of this sort. A matrix is a function $$A: \{1, 2, 3, .... , m\}\times \{1, 2, 3, .... , n\}\rightarrow\mathbb{R}.$$
So an matrix of higher dimensions is just a function from a cartesian product of finite integer sequences to $\mathbb{R}$.
The usage of $\mathbb{R}$; is pro-forma. You can use any set of objects in its stead.
When you say "think of a vector as a function", shouldn't that be a covector? At best I could see treating a vector as a function from $\{1,2,3,....n\}\rightarrow \mathbb{R}^n$, not to $\mathbb{R}$. – Robert Mastragostino Jun 8 '12 at 13:15