Kronecker delta

The Kronecker delta is a function of two indices that equals 1 when they are equal and 0 otherwise. Its entries form the identity matrix.

Identity matrix: A square matrix with ones on the main diagonal and zeros elsewhere; multiplying by it leaves a vector unchanged. Its entries are precisely the Kronecker delta values.
Matrix multiplication: An operation that combines rows of one matrix with columns of another to produce a new matrix. The identity entries given by the delta leave vectors and matrices unchanged under multiplication.
Orthonormal basis: A basis whose vectors are mutually orthogonal and each have unit length. Its basis-vector inner products are encoded by delta values.
Kronecker delta with multi-indices: A delta defined for tuples of indices, equal to one when the tuples match and zero otherwise. It extends equality testing from single indices to compound labels.
Index notation: A notation system that names components of vectors, matrices, and tensors with indices. The delta’s two indices specify which components match.
Tensor contraction: An operation that sums over paired tensor indices to reduce tensor order. A delta in a contraction identifies the paired indices.
Fourier series: A representation of a periodic function as a sum of sinusoidal modes. Orthogonality of its discrete modes yields Kronecker-delta relations.
Generalized Kronecker delta: An antisymmetric symbol defined by determinants of ordinary Kronecker deltas. It builds a higher-index object from the ordinary two-index delta.
Einstein notation: A convention in which repeated indices imply summation over their values. Repeated-index sums make the delta act as an index selector.
Levi-Civita symbol: A completely antisymmetric symbol whose components encode orientation and permutations of indices. Products of these symbols can be expressed using Kronecker deltas.

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