Kronecker Delta — Definition, Formula & Examples

The Kronecker delta is a function of two indices that equals 1 when the indices are equal and 0 when they are not. It provides a compact way to write the entries of the identity matrix and appears frequently in summation formulas throughout linear algebra and calculus.

The Kronecker delta, denoted $\delta_{ij}$ , is defined for integers $i$ and $j$ as $\delta_{ij} = 1$ if $i = j$ and $\delta_{ij} = 0$ if $i \neq j$ . Equivalently, for an $n \times n$ identity matrix $I_n$ , the entry in row $i$ and column $j$ is $\delta_{ij}$ .

Key Formula

\delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}

Where:

$i$ = First index (typically a row index)
$j$ = Second index (typically a column index)
$\delta_{ij}$ = Equals 1 or 0 depending on whether the indices match

How It Works

Think of

\delta_{ij}

as a filter: in a summation, it "selects" only the term where the two indices match and zeros out everything else. For instance,

\sum_{j=1}^{n} \delta_{ij} \, a_j = a_i

, because every term vanishes except the one where

j = i

. This sifting property makes the Kronecker delta essential for writing matrix equations in index notation. When you see

\delta_{ij}

in a formula, you can replace it with the corresponding entry of the identity matrix.

Worked Example

Problem: Use the Kronecker delta to evaluate the sum

\sum_{j=1}^{3} \delta_{2j} \, x_j

where

x_1 = 5

x_2 = 8

x_3 = -3

Expand the sum: Write out each term of the summation for j = 1, 2, 3.

\delta_{2,1}\,x_1 + \delta_{2,2}\,x_2 + \delta_{2,3}\,x_3

Apply the definition: Only

\delta_{2,2} = 1

; the other two deltas equal 0 because the indices differ.

0 \cdot 5 + 1 \cdot 8 + 0 \cdot (-3) = 8

Answer: The sum equals

8

, which is

x_2

. This demonstrates the sifting property: the Kronecker delta picked out the component whose index matched

i = 2

Why It Matters

The Kronecker delta is the standard way to express identity-matrix entries and orthonormality conditions in index notation. In physics and engineering, it appears in tensor algebra, Einstein summation convention, and the definition of orthonormal bases. Mastering it streamlines work in any course that uses component-wise matrix or vector formulas.

Common Mistakes

Mistake: Confusing the Kronecker delta with the Dirac delta function.

Correction: The Kronecker delta

\delta_{ij}

is defined on discrete integer indices and takes values 0 or 1. The Dirac delta

\delta(x)

is a distribution defined on continuous variables. They serve analogous sifting roles but in different mathematical settings.