A bilinear form takes two vectors and returns a scalar, linearly in each argument:

$$B:V\times V\to\mathbb R.$$

For a chosen basis, it can be represented by a matrix $B$ so that

$$B(\mathbf u,\mathbf v)=\mathbf u^T B\mathbf v.$$

More than a matrix

The matrix is a coordinate representation. If the basis changes, the numerical entries change even though the bilinear form itself is the same mathematical object.

The Euclidean dot product is one familiar bilinear form. Other bilinear forms can measure different geometric or physical relationships.

Second-order tensors

A second-order tensor can be understood as a multilinear object that, depending on its type, combines vectors and covectors to produce vectors, covectors or scalars.

In Euclidean engineering contexts, many second-order tensors are represented by matrices and act linearly on vectors:

$$\mathbf y=T\mathbf x.$$

Examples include conductivity, stress and inertia tensors.

Rank and order

Tensor order counts the number of vector/covector input slots involved in the multilinear object. It should not be confused with matrix rank.

Thinking of tensors by how they transform and act is more general than thinking of them merely as multidimensional arrays.