Kronecker's dalta (definition and application examples)

Last updated: Jan. 3, 2019
  Kronecker's delta is a function on $i, j = 1,2, \cdots, n$ defined as
Kronecker's dalta definition
Simple examples and its applications are described below.
 
Examples (n=3)
  The definition of Kronecker's delta is
, where $i, j = 1,2, \cdots, n$. For $n=3$
Kronecker's delta n=3
$$ \tag{1} $$
Identity matrix
  Let $A$ be the matrix whose elements are Kronecker's delta.
, where $i, j = 1,2,3$. By $(1)$, each element is specifically
In the matrix form of a matrix, we have
. We see that the matrix whose elements are equal to the Kronecker's delta is the identity matrix.
  Let $\mathbf{a}$ be an arbitrary three-dimensional vector,
From $(1)$, we have
In sumarry,
This relation is expressed as
which indicates that an arbitrary vector is unchanged by operating the identity matrix.
  Similarly, let B be an arbitrary 3x3 matrix defined as
. Using $(1)$, we have
, where $i,k=1,2,3$. This relation is expressed as
which indicates that an arbitrary matrix is unchanged by operating the identity matrix.
Orthonormal basis
Let $\{ \mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3} \}$ be an orthonormal basis on three dimensional space.
Using $(1)$ these can be represented collectively as
$$ \tag{2} $$ Thus, the definition of orthonormal basis can be expressed using Kronecker's delta.
Inner product
  Let $A$ and $B$ be arbitrary three-dimensional vectors $A$ and $B$. These can be expressed as
by an orthonormal basis $\{ \mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3} \}$. From $(2)$ and this, the inner product of $\mathbf{a}$ and $\mathbf{b}$ can be expressed using Kronecker's delta as
.
  Using $(1)$, we see that the right-hand side is equal to the the standard inner product,
. Thus the standard inner product can be expressed by Kronecker's delta.
Trace
Let $B$ be an arbitrary 3x3 matrix. By multiplying each component of $B$ by Kronecker's delta and summing it over all the components, it becomes equal to the trace of $B$.
.
  Although the above properties were proved only in the case of three dimensions, it is also true in the case of arbitrary finite dimensions.