A **bilinear function** (or *bilinear form*) is a function that’s bilinear for all arguments, which can be scalar or vector (Vinberg, 2003; Haddon, 2000). In other words, it is a linear function of x for every fixed y-value *and* a linear function of y for every x-value (Shilov & Silverman, 1963).

## Bilinear Function Formula and Examples

The function

is bilinear for every argument on the space ℂ[a, b], where:

- ℂ = the complex plane,
- [a, b] = closed interval from a to b.

A bilinear form can also be defined in terms of matrices. Every bilinear function on ℝ (the reals) has the form

(x, y) = x^{t}*A*y

for some n x n matrix *A*.

A couple of specific bilinear function examples (Karageorgis, 2020):

- An inner product on real-numbered vector space V; This bilinear form is positive definite and symmetric (its variables are unchanged under any permutation; In other words, if you switch out two of the variables, you end up with the same function),
- The dot product on ℝ is a bilinear function.

## Related Functions

A **symmetric bilinear function** is where f(u, v) = f(v, u) for all u and v. **Multilinear functions** are a generalization of bilinear functions; generally speaking, differential forms are alternating multilinear functions (Harvard, 2017).

## References

Haddon, J. 2000. Shape Representation and Bilinear Problems in Computer Vision. University of California, Berkeley.

Harvard University. (2017). Math 23b Problem Session 2 Solutions. Retrieved November 13, 2020 from: https://canvas.harvard.edu/files/3585874/download?download_frd=1&verifier=PeG9X5xVgZSQAsPJKN8X5xT8i5Gv3snwr3Rvbf9h

Karageorgis, P. (2020). Chapter 3. Bilinear forms; Lecture notes for MA1212. Retrieved November 13, 2020 from: https://www.maths.tcd.ie/~pete/ma1212/chapter3.pdf

Laetsch, T. (2012). Bilinear and Quadratic Exercises, p. 1.

Shilov, G. & Silverman, R. (1963). An Introduction to the Theory of Linear Spaces. Prentice-Hall.

Vinberg, E. (2003). A Course in Algebra. American Mathematical Society.