Outer Product

Relevant parts to questions...

• Outer product: multiply all components index-by-index; $C_{ik\ell m}=A_{ik}{B}_{\ell m}$.
• Rank of product = sum of ranks of the factors.
• Structures concatenate: a covariant factor and a mixed factor give a covariant-then-mixed product.
• Non-commutative in general: $A_{ik}{B}_{\ell m}\ne B_{ik}{A}_{\ell m}$.

The Outer Product (or tensor product, written $\otimes$) combines two tensors into a higher-rank tensor by multiplying their components index by index. For two rank-2 covariant tensors:

$$\boxed{\mathbf{C}=\mathbf{A}\otimes \mathbf{B},C_{ik\ell m}=A_{ik}{B}_{\ell m}}$$

The result is a rank-4 tensor. No summation - every combination of indices produces a distinct component.

Why the Outer Product is a Tensor

Each factor transforms independently, so the combined transformation simply stacks the $L$-matrices:

$$C_{ik\ell m}^{\prime }=A_{ik}^{\prime }{B}_{\ell m}^{\prime }=(L_{i^{\prime }}^n{L}_{k^{\prime }}^p{A}_{np})(L_{{\ell }^{\prime }}^r{L}_{m^{\prime }}^s{B}_{rs})=L_{i^{\prime }}^n{L}_{k^{\prime }}^p{L}_{{\ell }^{\prime }}^r{L}_{m^{\prime }}^s{C}_{nprs}$$

which is the transformation law for a rank-4 covariant tensor. One $L$ per free index, as always (see Tensor Transformation Rule).

Structure Concatenation

Unlike addition, the outer product does not require matching structures. The result’s structure is the concatenation of the factors’ structures:

• $A_{ij}\otimes B_k^{\cdot \ell }=C_{ijk}^{\cdot \cdot \cdot \ell }$ (covariant + mixed → mixed)
• $A_i^{\cdot j}\otimes B_k^{\cdot \ell }=C_{i\cdot k}^{\cdot j\cdot \ell }$
• $D_{ij}^{\cdot \cdot k}\otimes E_{\cdot m}^{\ell \cdot n}=F_{ij\cdot \cdot m}^{\cdot \cdot k\ell \cdot n}$

Properties

• Rank addition::$\text{rank}(A\otimes B)=\text{rank}(A)+\text{rank}(B)$.
• Non-commutative::$A_{ik}{B}_{\ell m}\ne B_{ik}{A}_{\ell m}$ in general - the positions of indices are swapped, which matters.
• Associative::$(A\otimes B)\otimes C=A\otimes (B\otimes C)$.
• Distributes over addition::$A\otimes (B+C)=A\otimes B+A\otimes C$.

Visualisation

For rank-1 tensors (vectors), the outer product builds a matrix:

$${\mathbf{v}}_1\otimes {\mathbf{v}}_2=\left(\begin{array}{c}2\\ 0\end{array}\right)\otimes \left(\begin{array}{cc}1& 7\end{array}\right)=\left(\begin{array}{cc}2& 14\\ 0& 0\end{array}\right)$$

For two matrices, the outer product is the Kronecker product, producing a block structure.

Applications

1. Building higher-rank tensors from simpler ones - e.g. the stress-energy tensor from products of velocity components.
2. Combining tensors of mismatched structure where addition would fail.
3. Setting up a contraction - form the outer product, then Contraction over chosen index pairs to form an inner product.

Outer product of different-rank tensors

Let $A_{ijk}$ be rank-3 covariant and $B^{\ell m}$ rank-2 contravariant. Their outer product

$C_{ijk}^{\cdot \cdot \cdot \ell m}=A_{ijk}{B}^{\ell m}$

is a rank-5 tensor with three covariant and two contravariant indices. $\boxed{\text{rank}=3+2=5}$.

Multiply every pair of components; rank adds; structures concatenate.