# Restricted bilinear map

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.

Thanks.

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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.