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A metabolic bilinear form is a nondegenerate symmetric bilinear form $B$ on a finite-dimensional vector space $V$ possessing a vector subspace $L$ withSuch an $L$ is a metabolizer. Equivalently, it is a totally isotropic vector subspace of half the dimension: $B$ vanishes on $L\times L$ and $\dim V=2\dim L$. Nondegeneracy gives $\dim L^{\perp }=\dim V-\dim L$, proving the equivalence.

The symmetric matrix $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ is metabolic: the line spanned by $(1,0)$ is its own bilinear orthogonal complement. Over a field of characteristic different from two, every metabolic symmetric bilinear form is a sum of such hyperbolic planes. In characteristic two, metabolic need not mean alternating: over ${\mathbb{F}}_2$, the identity matrix of size two has metabolizer spanned by $(1,1)$ but has $B((1,0),(1,0))=1$.

An integral Seifert form on a free abelian group $L$ of rank $2g$ is metabolic if there is a rank-$g$ direct summand $M\subset L$ on which the entire bilinear form vanishes:Over a field, replace the direct summand by a $g$-dimensional vector subspace. Such a half-rank vector subspace is a metabolizer. In a basis starting with a basis of the metabolizer, the Seifert matrix has a zero upper-left $g$-by-$g$ block.

The knot-theoretic nonsingularity condition concerns the matrix $V-V^T$, which defines a unimodular intersection pairing; $V$ itself need not be an invertible matrix. For example,has a metabolizer spanned by the first basis vector and has $\det (V-V^T)=1$, despite $\det V=0$. Requiring $V$ itself to be nonsingular would exclude this legitimate metabolic example. Algebraic concordance of knots identifies two forms when their difference is metabolic.