In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars).
In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: The dot product on
However, the matrices of a bilinear form on different bases are all congruent.
Every bilinear form B on V defines a pair of linear maps from V to its dual space V∗.
Define B1, B2: V → V∗ by This is often denoted as where the dot ( ⋅ ) indicates the slot into which the argument for the resulting linear functional is to be placed (see Currying).
For a finite-dimensional vector space V, if either of B1 or B2 is an isomorphism, then both are, and the bilinear form B is said to be nondegenerate.
More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with some other element: The corresponding notion for a module over a commutative ring is that a bilinear form is unimodular if V → V∗ is an isomorphism.
Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular.
For example, over the integers, the pairing B(x, y) = 2xy is nondegenerate but not unimodular, as the induced map from V = Z to V∗ = Z is multiplication by 2.
Given B one can define the transpose of B to be the bilinear form given by The left radical and right radical of the form B are the kernels of B1 and B2 respectively;[2] they are the vectors orthogonal to the whole space on the left and on the right.
If V is finite-dimensional then, relative to some basis for V, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero.
For a module over a commutative ring, a unimodular form is one for which the determinant of the associate matrix is a unit (for example 1), hence the term; note that a form whose matrix determinant is non-zero but not a unit will be nondegenerate but not unimodular, for example B(x, y) = 2xy over the integers.
A bilinear form is symmetric (respectively skew-symmetric) if and only if its coordinate matrix (relative to any basis) is symmetric (respectively skew-symmetric).
A bilinear form is alternating if and only if its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when char(K) ≠ 2).
A bilinear form is symmetric if and only if the maps B1, B2: V → V∗ are equal, and skew-symmetric if and only if they are negatives of one another.
If char(K) ≠ 2 then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows
A bilinear form B is reflexive if and only if it is either symmetric or alternating.
In a reflexive space the left and right radicals agree and are termed the kernel or the radical of the bilinear form: the subspace of all vectors orthogonal with every other vector.
The radical is always a subspace of V. It is trivial if and only if the matrix A is nonsingular, and thus if and only if the bilinear form is nondegenerate.
Definition: A bilinear form on a normed vector space (V, ‖⋅‖) is bounded, if there is a constant C such that for all u, v ∈ V,
Definition: A bilinear form on a normed vector space (V, ‖⋅‖) is elliptic, or coercive, if there is a constant c > 0 such that for all u ∈ V,
By the universal property of the tensor product, there is a canonical correspondence between bilinear forms on V and linear maps V ⊗ V → K. If B is a bilinear form on V the corresponding linear map is given by In the other direction, if F : V ⊗ V → K is a linear map the corresponding bilinear form is given by composing F with the bilinear map V × V → V ⊗ V that sends (v, w) to v⊗w.
The set of all linear maps V ⊗ V → K is the dual space of V ⊗ V, so bilinear forms may be thought of as elements of (V ⊗ V)∗ which (when V is finite-dimensional) is canonically isomorphic to V∗ ⊗ V∗.
Likewise, symmetric bilinear forms may be thought of as elements of (Sym2V)* (dual of the second symmetric power of V) and alternating bilinear forms as elements of (Λ2V)∗ ≃ Λ2V∗ (the second exterior power of V∗).
For modules (instead of vector spaces), just as how a nondegenerate form is weaker than a unimodular form, a nondegenerate pairing is a weaker notion than a perfect pairing.
[6] To define them he uses diagonal matrices Aij having only +1 or −1 for non-zero elements.
is called the real symmetric case and labeled R(p, q), where p + q = n. Then he articulates the connection to traditional terminology:[7] Some of the real symmetric cases are very important.
The positive definite case R(n, 0) is called Euclidean space, while the case of a single minus, R(n−1, 1) is called Lorentzian space.
Given a ring R and a right R-module M and its dual module M∗, a mapping B : M∗ × M → R is called a bilinear form if for all u, v ∈ M∗, all x, y ∈ M and all α, β ∈ R. The mapping ⟨⋅,⋅⟩ : M∗ × M → R : (u, x) ↦ u(x) is known as the natural pairing, also called the canonical bilinear form on M∗ × M.[8] A linear map S : M∗ → M∗ : u ↦ S(u) induces the bilinear form B : M∗ × M → R : (u, x) ↦ ⟨S(u), x⟩, and a linear map T : M → M : x ↦ T(x) induces the bilinear form B : M∗ × M → R : (u, x) ↦ ⟨u, T(x)⟩.
Conversely, a bilinear form B : M∗ × M → R induces the R-linear maps S : M∗ → M∗ : u ↦ (x ↦ B(u, x)) and T′ : M → M∗∗ : x ↦ (u ↦ B(u, x)).