A question about symmetric bilinear forms

Question

If b is an indefinite symmetric bilinear form is it nondegenerate?
And conversely if b is nondegenerate is it positive/negative definite or indefinite?
How can i start to prove this? Note:Edited and changed the question

Answer 1

$b$ is indefinite iff its matrix representation $B$ has both positive and negative eigenvalues; $b$ is nondegenerate iff $B$ has no zero eigenvalues. So,

• an indefinite $b$ can be degenerate (e.g. $B=\operatorname{diag}(1,0,-1)$);
• a nondegenerate $b$ can be positive definite, negative definite or indefinite (examples: consider $B=1,\ B=-1$ and $B=\operatorname{diag}(1,-1)$.)