# Why is there no guarantee that a linear combination of generic nondegenerate bilinear forms is still non degenerate?

Why is there no guarantee that a linear combination of generic nondegenerate bilinear forms is still non degenerate? Is there an example for that?

From here there is a claim that a linear combination of generic nondegenerate bilinear forms is still non degenerate if these nondegenerate bilinear forms are positive definite. Is there any reference to illustrate this claim or can you give a proof for this claim.

## 1 Answer

Why is there no guarantee that a linear combination of generic nondegenerate bilinear forms is still non degenerate? Is there an example for that?

$$I$$ is non-degenerate, but $$0=I-I$$ is degenerate.

From here there is a claim that a linear combination of generic nondegenerate bilinear forms is still non degenerate if these nondegenerate bilinear forms are positive definite.

It's false. What is true is that a positive combination of positive definite bilinear forms is positive definite.

• How about a positive linear combination of generic non degenerate complex bilinear forms? Is that still possible to be degenerate? Oct 31, 2018 at 19:21
• @Danny $0=I+(-I)$ is a positive combination of non-degenerate bilinear forms. Oct 31, 2018 at 19:23