\delta_{ij}

δᵢⱼ — Kronecker delta

Read "delta i j". It equals 1 when i and j are the same number and 0 when they differ. It is an if-statement written as a number.

$\delta_{ij} = 1$ if $i = j$ , and $0$ otherwise. It lets one formula cover two cases. In the softmax Jacobian, $p_i(\delta_{ij} - p_j)$ becomes $p_i(1 - p_i)$ when $i = j$ and $-p_i p_j$ when $i \ne j$ .

Written as a table over all $i$ and $j$ , it is exactly the identity matrix.

Example

\delta_{22} = 1, \quad \delta_{23} = 0

Same index gives 1; different indices give 0.

Where the course uses it

4.3 Logits, Softmax, and Cross-Entropy · Module 4, How a Model Learns From Datafirst use

Related terms

I — the identity matrixRead "I" or "the identity". It is a square matrix with 1s on the diagonal and 0s elsewhere. Multiplying by it leaves a vector unchanged.
Cases — an if/elseA large left brace with several lines is an if/else written vertically. Read each line as "this value, if this condition holds" and use the line whose condition is true.