Tensors versus Matrices: Differences

A matrix is a two-dimensional grid of size n×m that contains numbers: you can add and subtract matrices of the same size, multiply one matrix with another as long as the sizes are compatible((n×m)×(m×p)=n×p)((n×m)×(m×p)=n×p), and multiply an entire matrix by a constant.

A vector is a matrix with just one row or column (but see below), vector has direction, but tensor has no direction.

A tensor is often thought of as a generalized matrix. That is, it could be

• a 1-D matrix, like a vector, which is actually such a tensor,
• a 3-D matrix (something like a cube of numbers),
• a 0-D matrix (a single number), or
• a higher dimensional structure that is harder to visualize.

The dimension of the tensor is called its rank.

Any rank-2 tensor can be represented as a matrix, but not every matrix is really a rank-2 tensor. The numerical values of a tensor’s matrix representation depend on what transformation rules have been applied to the entire system.