Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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

Suppose $\phi$ is a bilinear form on the vector space $V$.

What does it mean (perhaps in matrix form) for $\phi|_X$ to be non-singular, where $X\leq V$?

This question is probably elementary, but I want to check that I have the right idea.


share|cite|improve this question

I suppose non-singular means non-degenerate: there are no nonzero vectors $x$ such that $B(x,y)=0$ for all vectors $y$ (both $x,y$ in the subspace in your case). Then if for a basis $(b_1,\ldots,b_n)$ of $V$ the matrix $M$ is defined by $M_{i,j}=B(b_j,b_i)$, and if $J$ is a $n\times m$ matrix (with $m=\dim X$) whose columns express a basis of the subspace $X$ in that basis, then the condition for $X$ to be non-degenerate is that the $m\times m$ matrix $J^\top\cdot M\cdot J$ is invertible.

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.