Tensor Addition

Relevant parts to questions...

• Tensors with the same rank and index structure can be added componentwise.
• Tensors with different ranks or different index positions cannot.
• The proof is always the same: factor the common $L$-matrices out of the sum.

Tensor Addition is defined componentwise. Given two tensors $A$ and $B$ of the same rank and same index structure (covariant, contravariant, or mixed in the same way), their sum is the tensor whose components are

$$\boxed{C_{ik}=A_{ik}+B_{ik}}$$

(and analogously for any index arrangement). The result is a tensor of the same rank and structure.

Why the Sum is a Tensor

Because both $A$ and $B$ transform with the same pattern of $L$-matrices, the transformation matrix factors out of the sum:

$$C_{ik}^{\prime }=A_{ik}^{\prime }+B_{ik}^{\prime }=L_{i^{\prime }}^m{L}_{k^{\prime }}^n{A}_{mn}+L_{i^{\prime }}^m{L}_{k^{\prime }}^n{B}_{mn}=L_{i^{\prime }}^m{L}_{k^{\prime }}^n(A_{mn}+B_{mn})=L_{i^{\prime }}^m{L}_{k^{\prime }}^n{C}_{mn}$$

So $C_{ik}$ obeys the covariant rank-2 transformation law. The identical argument - extracting the common $L$s - works for any matching index structure and any rank.

When Addition Fails

Different structures don't add

$A_{ik}+B^{ik}$ is not a tensor: one term transforms with $L_{i^{\prime }}^m{L}_{k^{\prime }}^n$, the other with $L_m^{i^{\prime }}{L}_n^{k^{\prime }}$, and those matrices cannot be factored out. The sum has no tensor transformation law.

Similarly, you cannot add tensors of different ranks (e.g. $A_{ik}+B_k$) - the dimensional shape itself disagrees.

Properties

• Commutative and associative::$A+B=B+A$ and $(A+B)+C=A+(B+C)$, directly from scalar arithmetic on each component.
• Cartesian collapse::all flavours coincide in Cartesian coordinates, so the structure-matching restriction quietly disappears there.
• Generalises to many summands::$P_{ijk}=A_{ijk}+B_{ijk}+C_{ijk}$ for any number of terms, provided every term shares rank and structure.

Applications

1. Building new tensors of the same shape, by linear combination::e.g. the stress tensor as a sum of contributions.
2. Decompositions into structurally simpler parts, such as the symmetric/antisymmetric split (see Symmetry and Antisymmetry).
3. Cartesian matrix addition is the special case of rank-2 tensor addition in an orthonormal basis.

Ordinary matrix addition

Let $M_{ij}=\left(\begin{array}{ccc}0& -2& -3\\ 1& 2& 5\\ 1& 3& 4\end{array}\right)$ and $N_{ij}=\left(\begin{array}{ccc}7& 8& 9\\ 6& 5& 4\\ 1& 2& 3\end{array}\right)$. The sum is

$[C_{ij}]=[M_{ij}]+[N_{ij}]=\boxed{\left(\begin{array}{ccc}7& 6& 6\\ 7& 7& 9\\ 2& 5& 7\end{array}\right)}$. The tensor character is preserved automatically - it is just entry-wise arithmetic.

Same rank + same structure → can add; factor out the $L$s.