Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
A bilinear form is continuous, so nondegenerate $\leftrightarrow$ positive or negative definite. EDIT:I assumed finite dimensions. – tst Jan 5 '13 at 14:09
But I don't use continuity here.I have a b bilinear form on V (that is finite dimensional real vector space ). – Serkan Yaray Jan 5 '13 at 14:21
I don't think that a bilinear form in finite dimensions can be discontinuous. – tst Jan 5 '13 at 14:27
The point is that a bilinear form in FD is actually a matrix, let it be $B$. Let $b$ be the largest entry of the matrix and $B_m$ the matrix with all entries equal to $b$. Then obviously $||B||\le ||B_m||$ but we already know that $||B_m||$ is finite, so $||B||$ is finite also, so $B$ is bounded thus continuous. – tst Jan 5 '13 at 14:31
The examples of the zero matrix (indefinite but degenerate), the identity matrix (nondegenerate and positive definite) and the diagonal matrix $(1,-1)$ (nondegenerate and indefinite) resolve both questions in the negative. – whuber Jan 5 '13 at 18:21

$b$ 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 $B=\operatorname{diag}(1,-1)$.)
share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.