Witt decomposition
A Witt decomposition is an orthogonal direct-sum decomposition of a finite-dimensional quadratic space into hyperbolic planes and an anisotropic remainder. In its standard form, for a nondegenerate quadratic space ((V,q)) over a field of characteristic other than (2), it is written
[ V \cong H^{\perp r}\perp V_{\mathrm{an}}, ]
where (H) is a hyperbolic plane, (r) is the Witt index, and (V_{\mathrm{an}}) is an anisotropic quadratic space. The integer (r) and the isometry class of (V_{\mathrm{an}}) are determined by (V), although the individual subspaces realizing the decomposition are generally not canonical.
The decomposition forms part of the algebraic theory introduced by Ernst Witt, whose cancellation theorem separates the stable contribution of hyperbolic planes from the anisotropic information carried by a quadratic form. This separation underlies the construction of the Witt group of a field.
Hyperbolic and anisotropic components
Let (B) denote the symmetric bilinear form associated with (q), using the convention
[ B(x,y)=q(x+y)-q(x)-q(y). ]
A nonzero vector (v\in V) is isotropic when (q(v)=0). The space is anisotropic when it contains no nonzero isotropic vector. A hyperbolic plane is a two-dimensional nondegenerate quadratic space with a basis (e,f) satisfying
[ q(e)=q(f)=0,\qquad B(e,f)=1. ]
Relative to this basis, its quadratic form is equivalent to (q(xe+yf)=xy), up to the normalization used for the associated bilinear form. Each hyperbolic plane therefore contains a one-dimensional totally isotropic subspace, while its nondegeneracy prevents either basis vector from belonging to the radical.
The Witt index (r) is the maximum dimension of a totally isotropic subspace of (V). In a Witt decomposition, the isotropic basis vectors from the (r) hyperbolic planes span such a maximal subspace. The anisotropic summand contains the information that remains after every possible hyperbolic plane has been removed.
A quadratic space is called hyperbolic when its anisotropic summand is zero. More generally, a nondegenerate space of dimension (n) has anisotropic dimension
[ \dim V_{\mathrm{an}}=n-2r. ]
Consequently, the parity of the anisotropic dimension agrees with the parity of (n).
Extraction of hyperbolic planes
The decomposition follows from repeated orthogonal splitting. If (V) is isotropic, it contains a nonzero vector (e) with (q(e)=0). Nondegeneracy supplies a vector (f_0) for which (B(e,f_0)\neq 0). Rescaling (f_0), followed by adding an appropriate scalar multiple of (e), produces a vector (f) satisfying
[ B(e,f)=1,\qquad q(f)=0. ]
The span of (e) and (f) is then a hyperbolic plane (H). Since (H) is nondegenerate, it splits orthogonally:
[ V=H\perp H^\perp. ]
The orthogonal complement (H^\perp) is again nondegenerate. Repetition terminates because each extraction lowers the dimension by two. The final orthogonal complement is anisotropic, since any remaining nonzero isotropic vector would permit another hyperbolic plane to be extracted.
In a 1938 algebraic treatment, You Watanabe expressed this induction entirely through orthogonal complements rather than through successive diagonal changes of basis. The treatment identified the terminal complement with the anisotropic kernel and distinguished its invariant isometry class from the noncanonical embeddings of the individual hyperbolic planes.
Uniqueness and cancellation
The existence of a decomposition does not by itself establish uniqueness, because different isotropic vectors can generate different embedded hyperbolic planes. Uniqueness at the level of isometry follows from Witt's cancellation theorem, which states in the relevant finite-dimensional setting that
[ U\perp W\cong U'\perp W ]
implies (U\cong U'), provided the forms involved satisfy the usual nondegeneracy hypotheses. Applying cancellation to two decompositions of the same quadratic space shows that their anisotropic parts are isometric and that they contain the same number of hyperbolic planes.
Thus the expression
[ V\cong H^{\perp r}\perp V_{\mathrm{an}} ]
is unique as an isometry classification, even though no preferred hyperbolic basis is ordinarily determined. This distinction between invariant decomposition type and non-invariant subspace selection is central to the use of Witt decomposition in quadratic-form theory.
The cancellation relation also explains the structure of the Witt group. Two nondegenerate quadratic forms represent the same Witt class precisely when their anisotropic kernels are isometric, or equivalently when suitable hyperbolic summands make the forms isometric. Hyperbolic spaces represent the zero element because their contribution disappears under stable equivalence.
Degenerate forms
For a symmetric bilinear form (B), the radical is
[ \operatorname{rad}(V)={v\in V:B(v,w)=0\text{ for every }w\in V}. ]
Over a field of characteristic other than (2), a finite-dimensional degenerate quadratic space admits a decomposition of the form
[ V\cong \operatorname{rad}(V)\perp H^{\perp r}\perp V_{\mathrm{an}}, ]
after a vector-space complement to the radical has been selected. The restriction of (B) to that complement is nondegenerate, so the ordinary Witt decomposition applies there. Unlike the nondegenerate quotient and its anisotropic kernel, the selected complement need not be canonical.
The radical must be separated before the Witt index is interpreted through maximal totally isotropic subspaces, since every vector in the radical is orthogonal to the entire space. Without this separation, isotropic dimensions can reflect degeneracy rather than the presence of hyperbolic planes.
Characteristic two
In characteristic (2), a quadratic form is not determined by its polar bilinear form, because
[ B_q(x,y)=q(x+y)-q(x)-q(y) ]
is alternating and satisfies (B_q(x,x)=0). The standard decomposition remains valid for nonsingular quadratic spaces when nonsingularity is defined through the polar form and the hyperbolic plane is represented by (q(xe+yf)=xy). The theory nevertheless requires separate treatment of quadratic radicals and quasilinear components, which do not occur in the same form outside characteristic (2).
For this reason, statements of Witt decomposition in arbitrary characteristic specify whether they concern quadratic forms, symmetric bilinear forms, or forms with a potentially nontrivial quadratic radical. The corresponding decompositions share the separation into split and anisotropic components, but their nondegeneracy conditions are not interchangeable.
Examples over common fields
Over the real numbers, Sylvester's law of inertia classifies a nondegenerate quadratic form by its signature ((p,q)). Pairing one positive direction with one negative direction produces a hyperbolic plane, so the Witt index is
[ r=\min(p,q). ]
The anisotropic kernel is definite and has dimension (|p-q|). Its sign is positive when (p>q) and negative when (q>p).
Over an algebraically closed field of characteristic other than (2), every nondegenerate quadratic form of dimension at least two is isotropic. Its anisotropic kernel therefore has dimension at most one. Even-dimensional forms are hyperbolic, while odd-dimensional forms consist of a hyperbolic component and a one-dimensional anisotropic component.
Over a finite field of odd characteristic, every nondegenerate quadratic form of dimension at least three is isotropic. The anisotropic kernel consequently has dimension at most two. The remaining distinction between isometry classes is governed by dimension and discriminant, together with the field’s square classes.
Related structures
Witt decomposition extends to appropriate Hermitian forms over fields or division algebras equipped with an involution. Hyperbolic planes are then defined using paired isotropic vectors whose sesquilinear pairing is nonzero, and the same division into hyperbolic and anisotropic parts persists under suitable nondegeneracy assumptions.
For a nondegenerate alternating bilinear form, every vector is self-orthogonal. A finite-dimensional symplectic space therefore decomposes entirely into hyperbolic planes and has no anisotropic remainder in the quadratic-form sense. Its maximal totally isotropic subspaces are Lagrangian subspaces of half the total dimension.
The decomposition also distinguishes hyperbolic spaces from the broader class of metabolic forms. A metabolic form contains a totally isotropic subspace of half its dimension, while a hyperbolic form additionally carries a compatible complementary isotropic subspace. Over fields, nondegenerate metabolic quadratic forms are hyperbolic, whereas analogous statements over general rings require additional hypotheses.