They look a lot like matrices, and seem to work exactly like matrices. What is the difference between them? I have only worked with matrices, not tensors because I cant find a tutorial online but every time I have seen one they seem identical.
Example: The identity matrix is a diagonal matrix of 1's. If the coordinate system is in feet or inches, the diagonals are still 1's. So the identity matrix is a math concept that does not transform correctly (from coordinates of feet to coordinates of inches) to represent a physical entity. For the same matrix to represent a tensor, it would have to be defined in a way that its diagonal 1's in the coordinates of feet would transform to either 12's or 1/12's diagonal elements in coordinates of inches (there are covarient and contravarient tensors)
Ok. I retract my statement and will stay out of this discussion.While I agree that transformation properties of tensors are important, I think the unit matrix is not a very illuminating (and somewhat misleading) example. In particular, consider the (1,1)-tensor ##\delta^\alpha_\beta## such that ##\delta^\alpha_\beta V^\beta = V^\alpha##, where ##V## is a vector. This tensor will be represented by the unit matrix in all frames (the unit matrix is a transformation from the vector space of column matrices to itself and therefore naturally represents a (1,1)-tensor, you can fiddle around to make a square matrix represent an arbitrary rank-2 tensor, but I would say it is slightly less natural). The tensor transformation properties follow trivially from the chain rule.