The inner product in the context of tensors is a rank-contraction operation[0]. The IP you know and love contracts over the covariant and contravariant indicies

The inner product in the context of tensors is a rank-contraction operation[0]. The IP you know and love contracts over the covariant and contravariant indicies of two rank-1 tensors (vectors). You need a metric to do this so you can raise one of the contravariant indicies to be covariant. If your metric is the normal one (all 1's) then you just add the product of all coefficients. Though if you have the Minkowski metric (like in special relativty) then you negate the time coordinate (or all the space coordinates) and add. Here the matrix might be regarded as a rank 1,1 tensor (one covariant, one contravariant index) that you contract with the vector. [0] http://en.wikipedia.org/wiki/Tensor_contraction http://en.wikipedia.org/wiki/Tensor_contraction