In mathematics, what is an $N \times N \times N$ matrix? In mathematics, what is an $N \times N \times N$ matrix? I think this is a tensor but definitions of tensors that I have read are so overly complicated and verbose that I have trouble understanding them.
 A: Tensors in general are multilinear coordinate-free objects that can be represented with respect to some basis by a multi-dimensional arrays indexed appropriately. Just like you can represent a bilinear form $B \colon V \times V \rightarrow  \mathbb{F}$ when $V$ is an $n$-dimensional vector space (or a linear map $T \colon V \rightarrow V$) by a $n \times n$ matrix and use a matrix (together with a choice of basis) to define a bilinear form (or a linear map) on $V$, you can represent a multi-linear map $B' \colon V \times V \times V \rightarrow \mathbb{F}$ ("a tensor") by an $n \times n \times n$ dimensional array of scalars and use such an array to define a multi-linear map on $V$.
A: You can think of a rank three tensor as a three dimensional array. A matrix is a rank two tensor, or a two dimensional array. A vector then is a rank one tensor and scalar a rank zero.  This is a simplification of the subject of tensors, but it's useful to think of them as a generalization of matrices.
A: I'ts a tensor. But so are normal numbers, vectors, and matrices which can be considered 1X1 tensor, 1Xn tensor, mXn Tensor. All data structures of this form (mXnX...)are what are called Tensors.
A: You can think of tensors as multidimensional arrays. 
A: If you are having trouble with tensors, try this:
Imagine that you want to locate a point in 3-dimensional Cartesian plane. You need (x,y,z) coordinates to locate the point. Now imagine that each of these points has a numerical value. Then this three dimensional plane with these numerical value will constitute a matrix. If the length of x, y and z axes, each is N, then you get $N\times N\times N$ matrix.
