and from it again by “reduction” the tensor of the zero rank
A = A_{β}^{β} = A_{αβ}^{αβ}.
The proof that the result of reduction retains a truly tensorial
character, follows either from the representation of tensor according to
the generalisation of (12) in combination with (6) or out of the
generalisation of (13).
_Inner and mixed multiplication of Tensors._
This consists in the combination of outer multiplication with reduction.
Examples:—From the co-variant tensor of the second rank A_{μν} and the
contravariant tensor of the first rank B^{σ} we get by outer
multiplication the mixed tensor
D^{σ}_{μν} = A_{μν} B^{σ} .
Through reduction according to indices ν and σ (_i.e._, putting ν = σ),
the co-variant four vector
D_{μ} = D^{ν}_{μν} = A_{μν} B^{ν} is generated.
These we denote as the inner product of the tensor A_{μν} and B^{σ}.
Similarly we get from the tensors A_{μν} and B^{στ} through outer
multiplication and two-fold reduction the inner product A_{μν} B^{μν}.
Through outer multiplication and one-fold reduction we get out of A_{μν}
and B^{στ}, the mixed tensor of the second rank D^{τ}_{μ} = A_{μν}
B^{τν}. We can fitly call this operation a mixed one; for it is outer
with reference to the indices μ and τ and inner with respect to the
indices ν and σ.
We now prove a law, which will be often applicable for proving the
tensor-character of certain quantities. According to the above
representation, A_{μν} B^{μν} is a scalar, when A_{μν} and B^{στ} are
tensors. We also remark that when A_{μν} B^{μν} is an invariant for
every choice of the tensor B^{μν}, then A_{μν} has a tensorial
character.