Einsum Notation — Rules

The single master rule: einsum iterates over all index combinations, multiplies the corresponding elements, and sums over any index not in the output.

1. Basic Syntax

"inputs -> output", where inputs are comma-separated index strings for each operand.

2. Index Letters

Each letter represents one axis (dimension) of a tensor. The size of that dimension must be consistent wherever the letter appears.

3. Free Indices (output indices)

Any index that appears on the right side of -> is a free index — it survives into the output.

4. Contracted Indices (summed indices)

Any index that appears in the input(s) but not in the output is a dummy index — it gets summed over (contracted).

• "ij,jk->ik": j is dummy → summed → matrix multiply
• "ij->i": j is dummy → summed → row sums

5. Repeated Index Within One Operand → Diagonal

Same letter twice in one operand constrains to the diagonal along those axes.

• "ii->i": diagonal elements → vector
• "ii->": diagonal then sum → trace

6. Shared Index Across Operands → Alignment

Same letter in two operands: aligned along that dimension before multiplying.

• In output → element-wise multiply (kept)
• Not in output → multiply and sum (contracted)

7. The Optional Arrow (->)

With arrow: You fully control which indices appear in the output and in what order.

Without arrow (implicit): Include each index appearing exactly once (alphabetical order), sum any appearing more than once.

8. Batch / Spectator Indices

An index in multiple operands and in the output is a batch index — no summation, operation performed independently per value.

"bij,bjk->bik": b is batch → batched matrix multiply

9. Output Ordering

The order of indices in the output string determines the shape. This lets you transpose implicitly: "ij->ji"

Quick Reference

Links

• einsum.joelburget.com
• Einsum Puzzles (Colab)