Editing Tensor (section)

=== Raising or lowering an index ===
{{Main|Raising and lowering indices}}
When a vector space is equipped with a [[nondegenerate bilinear form]] (or ''[[metric tensor]]'' as it is often called in this context), operations can be defined that convert a contravariant (upper) index into a covariant (lower) index and vice versa.  A metric tensor is a (symmetric) ({{nowrap|0, 2)}}-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product.  This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index.  This operation is quite graphically known as ''lowering an index''.

Conversely, the inverse operation can be defined, and is called ''raising an index''.  This is equivalent to a similar contraction on the product with a {{nowrap|(2, 0)}}-tensor.  This ''inverse metric tensor'' has components that are the matrix inverse of those of the metric tensor.