General properties of tensors

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Tensors have the following general properties:

• If and are M/N tensors, so are , and for any and . Thus tensors of a given type form a vector space .
• If is a M/N tensor and is of type M'/N', then the outer product is a tensor of type M+M'/N+N'.
• For any tensor of type M/N, one can construct a tensor of type M-1/N-1 by contracting   an upper index with a lower index for example is constructed from [ or ]. Note however that there may be several ways of contracting and they give different tensors.
• Tensors can be symmetric  on pairs of upper or lower indices for example and , or anti- symmetric . However symmetric or anti- symmetric pairs of indices with one index up and the other down cannot be defined since the relationship need not hold in all frames.