Suffix Notation

Relevant parts to questions…

• Converting everything into suffix form by applying free and dummy indices.
• Using $\mathbf{a}\cdot \mathbf{b}=a_i{b}_i$, both ways.
• Using $\mathbf{AB}=A_{ik}+B_{kj}$ and $(A_{ij}{)}^T=A_{ji}$.

Suffix Notation, or Index Notation, is simply when write a sum without the $\Sigma$ symbol. This results in an expression with two types of index:

• Dummy indices::Indices that are iterated over within a sum, e.g., ${\sum }_{j=1}^3{a}_j{b}_j\to \boxed{a_j{b}_j}$ - arbitrary variable naming with no presence in result.
• Free indices::Indices that represent a component within a tensor, e.g., the $i$th component of a vector - consistent variable naming for comparison across vectors.

We can convert to this notation by introducing these two index types:

1. Introduce free index, like $i$: $(\mathbf{u}+(\mathbf{a}\cdot \mathbf{b})\mathbf{v}{)}_i=((\mathbf{a}\cdot \mathbf{a})(\mathbf{b}\cdot \mathbf{v})\mathbf{a}{)}_i:|\mathbf{a}|:=\sqrt{\mathbf{a}\cdot \mathbf{a}}$;
2. Distribute free index, for each vector.: $u_i+(\mathbf{a}\cdot \mathbf{b})v_i=(\mathbf{a}\cdot \mathbf{a})(\mathbf{b}\cdot \mathbf{v})a_i$;
3. Introduce dummy indices, for each scalar: $\boxed{u_i+a_j{b}_j{v}_i=a_k{a}_k{b}_{\ell }{v}_{\ell }{a}_i}$.