Throughout is a field and all vetor spcaes are over

**Definition 1**. Let be a -vector space. A **bilinear form** on is a map such that for every and we have

1)

2)

**Definition 2**. Let be a bilinear form on a vector space we say that is **non-degenerate** if implies If is a basis for then the matrix whose -entry is is called **the matrix of** with respect to that basis of

**Theorem**. Let be a finite dimensional vector space with a bilinear form Then is non-degenerate if and only if the matrix of is invertible.

*Proof*. Choose a basis for Let be the matrix of Then for some if and only if for all where is the -th coordinate of Thus if and only if for all if and only if for all Hence has a non-zero solution for if and only if there exists such that for all Thus is not invertible if and only if is not non-degenerate.

**Remark**. Since is invertible if and only if the transpose of is invertible, a similar argument to the above theorem gives us that is non-degenerate if and only if implies

**Definition 3**. A finite dimensional -algebra is called a **Frobenius algebra** if as a -vector space, has a non-degenerate bilinear form such that for all

**Example**. Let the algebra of matrices with entries in Then is a Frobenius -algebra.

*Proof*. Define by the trace of It is clear that is a bilinear form. To prove that is non-degenerate, suppose that for all So for all Let be the element of with -entry and anywhere else. Let be the -entry of Let and let Then and so This proves that is non-degenerate. Finally, for any we have