The Role of Matrices

What is a matrix? We consider matrices as elements of \(\mathbb K^{m \times n}\) . We have first to clarify a potential source of confusion. Matrices by themselves are not tensors in the sense of tensor calculus. They do not have a specific transformation rule, so, for example, the transformation property of a matrix representing a linear map is different to the transformation property of a matrix representing a given scalar product. Therefore we should always clarify how to use matrices in a given situation. Matrices seem to be very flexible objects, and we use this feature in mathematics and physics. We do not consider matrices only with scalars, but we also use matrices with entries other than scalars, for example, entries with vectors or covectors (linear forms), or even matrices. The most prominent property of all matrices is that we can add and, under certain conditions, multi- ply matrices. The multiplication rule seems at first sight quite complicated, but it turns out to be a reasonable and practical approach. In most cases, matrices are used as a representation of linear maps. In addition, the multiplication rule is justified by the composition of linear maps.