In the mathematical theory of tensors, they are defined as abstract mathematical entities or mathematical operators. As mathematical operators, they are used to operate on other tensors either to reduce their ranks or to increase them. Nevertheless, tensor ranking is simply a way to keep track the changes in direction of certain physical quantities like the electric field and magnetic field. This tracking system is very important in an anisotropic medium. On the other hand, in an isotropic medium, all tensors would share the same directional invariance properties and they can be reduced either into vectors or scalar physical quantities.


In tensor or vector analysis, tensor ranking can be done simply by the use of matrix algebra. In this sense, a zero rank tensor called a scalar is simply a tensor with one element. Tensors of rank one are called vectors, and rank two tensors are simply matrices. Ranks zero, one, and two tensors can all be classified as matrices. This generalization would define a scalar as a matrix containing one element. Vectors can be defined in two ways: column matrix or row matrix. Matrix remains as matrix with the same rules for matrix multiplication. By these rules, the product of a row matrix with a column matrix is a scalar, while the product of a column matrix with a row matrix is a matrix of the same order as the number of elements of the row and column matrix. For example, the product of 1xm and mx1 is a scalar. The product of mx1 and 1xm is a matrix of order mxm. The product of mx1 and 1xn is a matrix of order mxn.