There are also vector quantities with covariant indices. They are generally obtained from their corresponding objects with contravariant indices by the operation of lowering an index ; e.g., x ν ν = η η μ μ ν ν x μ μ , where η is the metric tensor. (The linked article also provides more information about what the operation of raising and lowering indices really is mathematically.) The inverse of this transformation is given by x μ μ = η η μ μ ν ν x ν ν , where, when viewed as matrices, η is the inverse of η μν. As it happens, η = η μν. This is referred to as raising an index. To transform a covariant vector A μ, first raise its index, then transform it according to the same rule as for contravariant 4 -vectors, then finally lower the index; A ′ ν ν = η η ρ ρ ν ν Λ Λ ρ ρ σ σ η η μ μ σ σ A μ μ .