Kronecker's dalta (definition and application examples)

  Kronecker's delta is a function on $i, j = 1,2, \cdots, n$ defined as Simple examples and its applications are described below.     The definition of Kronecker's delta is , where $i, j = 1,2, \cdots, n$. For $n=3$ $$ \tag{1} $$   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. 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.   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. 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 $B$. .