Source: /cirosantilli/matrix-representation-of-a-bilinear-form

= Matrix representation of a bilinear form

As usual, it is useful to think about how a <bilinear form> looks like in terms of <vectors> and <matrices>.

Unlike a <linear form>, which <matrix representation of a linear form>[was a vector], because it has two inputs, the bilinear form is represented by a matrix $M$ which encodes the value for each possible pair of <basis vectors>.

In terms of that <matrix>, the form $B(x,y)$ is then given by:
$$
B(x,y) = x^T M y
$$