In order to answer that question we first treat orthonormality. The following relation is required to hold:
i.e.
Therefore, only matrices with the following property are permitted
or
.
Mathematicians call these matrices orthogonal. From their nine
matrix elements only three real numbers are independent because
of the six constraint equations:
 |
for
and |
 |
for
 . |
With respect to matrx multiplication, the orthogonal matrices form a group
called
which
of course can not be Abelian, since we have found the
multiplication of matrices in general not to be commutative:
To verify the group property we consider first the product
of
two orthogonal matrices
with
and
with
and calculate:
meaning the product of two orthogonal matrices is again orthogonal.
Furthermore the associative law holds true as for every matrix multiplication:
Exactly one unit element exists, since the unit matrix
is orthogonal because from
it follows
that
:
unit element
:
for
all
For multiplication of a matrixfrom
the left or from the right with the unit matrix
one obtains again the old matrix.
And an unambiguously determined inverse exists for every orthogonal matrix
, namely
the transposed matrix. For this was precisely the orthogonality condition:
The necessary condition for the existence of an inverse
is fulfilled, since from
there follows for the determinant
With this, all group properties of orthogonal matrices are proven. From
the determinant we see furthermore that two
kinds of orthogonal matrices exist: those with determinant
,
the rotations, and those with determinant
.
The latter are just the reflections.
The defining equation for the orthogonal matrices
opens up a fully different view on our Kronecker symbol: If we include on
the left side a superfluous
with a further summation, we obtain:
.
If we regard the Kronecker symbol, because of its two indices, as a
second order tensor, we find on the left side
,
i.e. the nine elements of this tensor transformed to the new coordinate
system: for each index a transformation matrix. Thus the entire equation
means the invariance of the matrix elements under rotations and
reflections, i.e. the ones and zeros are unchanged in every coordinate
system and stay at the same position: The Kronecker symbol, symmetric
against interchanging the two indices, is from a higher point of view a "numerically
invariant tensor of second order". You will frequently encounter it later on
under this name.