Bilinear Forms and Orthogonality

A bilinear form (or inner product) on V is a bilinear function f : V × V →k. The ordered pair (V, f ) is called an inner product space. Bilinear Forms Of course, (kn, f ) is an inner product space if f is the familiar dot product f (u, v) = i uivi, where u = (u1, . . . , un)t and v = (v1, · · · , vn)t (the superscript t denotes transpose; remember that the elements of kn are n×1 column vectors). In terms of matrix multiplication, we have f (u, v) = utv. There are two types of bilinear forms of special interest.

Let f be a bilinear form on a vector space V over a field k, and let E = e1, . . . , en be a basis of V . Then an inner product matrix of f relative to E is A = [ f (ei, e j)]. Suppose that (V, f ) is an inner product space, e1, . . . , en is a basis of V , and A = [ f (ei, e j)] is the inner product matrix of f relative to E. If b = biei and c = ciei are vectors in V , then f (b, c) = f ( biei, ciei ) = i, j bi f (ei, e j)c j. If B and C denote the column vectors (b1, . . . , bn)t and (c1, . . . , cn)t, respectively, then this last equation can be written in matrix form: f (b, c) = Bt AC. Thus, an inner product matrix determines f completely. Bilinear Forms

If (V, f ) is an inner product space and W ⊆V is a subspace of V , then the left orthogonal complement of W is W ⊥L = {b ∈V : f (b, w) = 0 for all w ∈W}; the right orthogonal complement of W is W ⊥R = {c ∈V : f (w, c) = 0 for all w ∈W}. Moreover, W ⊥L = W ⊥R if f is either symmetric or alternating, in which case we write W ⊥. Let (V, f ) be an inner product space, and let A be the inner product matrix of f relative to a basis e1, . . . , en of V . We claim that b ∈V ⊥L if and only if b is a solution of the homogeneous system Atx = 0. If b ∈V ⊥L. then f (b, e j) = 0 for all j. In matrix terms, b = (b1, . . . , bn)t and Bt A = 0; transposing, b is a solution of the homogeneous system Atx = 0. The proof of the converse is left to the reader. A similar argument shows that c ∈V ⊥R if and only if c is a solution of the homogeneous system Ax = 0. Bilinear Forms

A hyperbolic plane over a field k is a two-dimensional vector space over k equipped with a nonzero alternating bilinear form. We have just seen that every two-dimensional alternating space (V, f ) in which f is not identically zero has an inner product matrix A = 0 1 −1 0 .

Let (V, f ) be a symmetric space, and let E = e1, . . . , en be a basis of V . If e1, . . . , en is an orthogonal basis of a symmetric space (V, f ), then V = ⟨e1⟩⊕· · ·⊕ ⟨en⟩is an orthogonal direct sum. In Corollary 9.76, we saw that if (V, f ) is a symmetric space with f nondegenerate, and if e1, . . . , en is a basis of V , then there exists a basis b1, . . . , bn of V with f (ei, b j) = δi j. If E is an orthonormal basis, then we can set bi = ei for all i.

Let V be an a vector space over a field k. A quadratic form is a function Q : V →k such that (i) Q(cv) = c2Q(v) for all v ∈V and c ∈k; (ii) the function g : V × V →k, defined by g(u, v) = Q(u + v) −Q(u) −Q(v), is a bilinear form.

Let (V, f ) be an inner product space, where V is a finite-dimensional vector space over a field k and f is a nondegenerate bilinear form. (i) If f is alternating, then Isom(V, f ) is called the symplectic group, and it is denoted by Sp(V, f ). (ii) If f is symmetric, then Isom(V, f ) is called the orthogonal group, and it is denoted by O(V, f ). Bilinear Forms As always, a choice of basis E of an n-dimensional vector space V over a field k gives an isomorphism µ: GL(V ) →GL(n, k), the group of all nonsingular n × n matrices over k. In particular, let (V, f ) be an alternating space with f nondegenerate, and let E = x1, y1, . . . , xm, ym be a symplectic basis of V (which exists, by Corollary 9.80); recall that n = dim(V ) is even; say, n = 2m. Denote the image of Sp(V, f ) by Sp(2m, k). Similarly, if (V, f ) is a symmetric space with f nondegenerate, and E is an orthogonal basis (which exists when k does not have characteristic 2, by Theorem 9.81), denote the image of O(V, f ) by O(n, f ). Let us find adjoints when the bilinear form is symmetric or alternating.

An R-algebra A is a graded R-algebra if there are R-submodules Ap, for p ≥0, such that (i) A = p≥0 Ap; (ii) For all p, q ≥0, if x ∈Ap and y ∈Aq, then xy ∈Ap+q; that is, Ap Aq ⊆Ap+q. An element x ∈Ap is called homogeneous of degree p. Notice that 0 is homogeneous of any degree, but that most elements in a graded ring are not homogeneous and, hence, have no degree. Note also that any product of homogeneous elements is itself homogeneous.

Let k be a field of characteristic ̸= 2, and let f be a bilinear form defined on a vector space V over k. Then there are unique bilinear forms fs and fa, where fs is symmetric and fa is alternating, such that f = fs + fa.

If (V, f ) is an inner product space and if A and A′ are inner product matrices of f relative to different bases of V , then there exists a nonzero a ∈k with det(A′) = a2 det(A). Consequently, A′ is nonsingular if and only if A is nonsingular.

Let (V, f ) be an inner product space, and let e1, . . . , en be a basis of V . Then f is nondegenerate if and only if f ( , e1), . . . , f ( , en) is a basis of the dual space V ∗(we call the latter the dual basis).

Let (V, f ) be an inner product space with f nondegenerate. If e1, . . . , en is a basis of V , then there exists a basis b1, . . . , bn of V with f (ei, b j) = δi j.

, there are b1, . . . , bn ∈V with gi = f ( , bi) for all i. Commutativity of the diagram gives f (ei, b j) = ev(ei, g j) = δi j. •

Let (V, f ) be an alternating space, where V is a vector space over a field k. If f is nondegenerate, then there is an orthogonal direct sum V = H1 ⊕· · · ⊕Hm, where each Hi is a hyperbolic plane.

Let (V, f ) be an alternating space in which f is nondegenerate,14 and let A be an inner product matrix for f (relative to some basis of V ). (i) There exists a symplectic basis x1, y1, . . . , xm, ym for V , and A is a 2m ×2m matrix for some m ≥1. (ii) A is congruent to a matrix direct sum of blocks of the form −1 , and the latter is congruent to I −I , where I is the m × m identity matrix. (iii) Every nonsingular skew-symmetric matrix A over a field k is congruent to a direct sum of 2 × 2 blocks −1 .

If f is a symmetric bilinear form on a vector space V over R of dimension m, then there is an orthogonal direct sum V = W+ ⊕W−⊕W0, where f |W+ is positive definite, f |W−is negative definite, and f |W0 is identically 0. Moreover, the dimensions of these three subspaces are uniquely determined by f .

Let (V, f ) be an inner product space, where f is a nondegenerate bilinear form, let E = e1, . . . , en be a basis of V , and let A be the inner product matrix relative to E. Then ϕ ∈GL(V ) is an isometry if and only if its matrix M = E[ϕ]E satisfies the equation Mt AM = A.

If (V, f ) is an inner product space whose bilinear form f is nondegenerate, then every linear transformation T : V →V has an adjoint.

Let (V, f ) be an inner product space whose bilinear form f is nondegenerate. If T : V →V is a linear transformation with adjoint T ∗, then T is an isometry if and only if T ∗T = 1V , in which case T ∗= T −1.

Let (V, f ) be an alternating space, where V is a 2m-dimensional vector space over a field k and f is nondegenerate, and let E be a symplectic basis ordered as x1, . . . , xm, y1, . . . , ym. The adjoint of a matrix B = P Q S T relative to E, partitioned into m × m blocks, is B∗= T t −Qt −St Pt . Remark. It follows that B ∈Sp(2m, k) if and only if B∗B = I. ◀

(i) Let (V, f ) be an inner product space, where V is a finite-dimensional vector space over a field k and f is a nondegenerate bilinear form. The stabilizer GL(V ) f of f under the action on kV ×V is Isom(V, f ). (ii) If g : V × V →k lies in the same orbit as f , then Isom(V, f ) and Isom(V, g) are isomorphic; in fact, they are conjugate subgroups of GL(V ).

shows that det(P) = 1 for P ∈Sp(2, k) [it is true, for all m ≥1, that Sp(2m, k) ≤SL(2m, k)]. 9.60 If A is an m × m matrix with At A = I, prove that A A is a symplectic matrix. Conclude, if k is a finite field of odd characteristic, that O(m, k) ≤Sp(2m, k). 9.61 Let (V, f ) be an alternating space with f nondegenerate. Prove that T ∈GL(V ) is an isometry [i.e., T ∈Sp(V, f )] if and only if, whenever E = x1, y1, . . . , xm, ym is a symplectic basis of V , then T (E) = T x1, T y1, . . . , T xm, T ym is also a symplectic basis of V . 9.6 GRADED ALGEBRAS We are now going to use tensor products of many modules in order to construct some useful rings. This topic is often called multilinear algebra. Throughout this section, R will denote a commutative ring.

(i) If V = k2 and its elements are viewed as column vectors, then det: V × V →k, given by a b , c d ↦det a c b d = ad −bc, is an example of an alternating bilinear form. More generally, define a function f : Cn × Cn →C by f (u, v) = j u jv j, where u = (u1, . . . , un)t, v = (v1, . . . , vn)t, and c denotes the complex conjugate of a complex number c. Such a function is not C-bilinear because f (u, cv) = c f (u, v) instead of cf (u, v). conjugate-symmetric forms are examples of sesquilinear forms; such forms can be constructed over any field k equipped with an automorphism of order 2 (to play the role of complex conjugation). ◀ Every bilinear form can be expressed in terms of symmetric and alternating bilinear forms.

(i) The polynomial ring A = R[x] is a graded R-algebra if we define Ap = {rx p : r ∈R}. The homogeneous elements are the monomials and, in contrast to ordinary usage, only monomials (including 0) have degrees. On the other hand, x p has degree p in both usages of the term degree. (ii) The polynomial ring A = R[x1, x2, . . . , xn] is a graded R-algebra if we define Ap = { rxe1 1 xe2 2 · · · xen n : r ∈R and ei = p } ; that is, Ap consists of all monomials of total degree p.