Tensor

Vectors and matrices are examples of more general objects called tensors. Tensors are defined via their transformation properties: suppose we have a set of numbers $v_{i}$, and we want to know how their values change under rotation of Cartesian axes. If the values in the new co-ordinate system $v'_{i}$ can be written

$v'_{i}=L_{ij}v_{j}$

where $L_{ij}$ are the elements of a rotation matrix then the $v_{i}$ are said to be the components of a rank one tensor. Similarly, the components of a rank two tensor satisfy

$a'_{ij}=L_{im}L_{jn}a_{mn}$

and for higher order tensors, we just keep adding more of the $L_{ij}$ rotation matrices. Scalars, vectors and matrices are rank zero, rank one and rank two tensors respectively.

See also

Suffix notation