PROBLEMS ON BILINEAR FORMS

(1) (a) Let K be a field with char(K) 6= 2, and let Q(x1, . . . , xn) ∈ K[x1, . . . , xn] be homogeneous of degree two. Show that 1 B(~x,~y) := Q(x + y , . . . , x + y ) − Q(x , . . . , x ) − Q(y , . . . , y )
2 1 1 n n 1 n 1 n defines a symmetric bilinear form on Kn. (Note: such Q are called quadratic forms.)

(b) Converseley, show that if B(~x,~y) is a symmetric bilinear form on Kn, then

Q(x1, . . . , xn) := B(~x,~x) gives a homogeneous degree two polynomial, and that B(~x,~y) is recovered by the proce- dure in part (a). Use this to give an alternative proof that if B(~x,~y) is a non-degenerate bilinear form on Kn, then there exists a non-zero vector so that B(~x,~x) 6= 0. (Recall this was the first step in our proof that symmetric bilinear forms can be diagonalized).

(2) Suppose that g(v, w) is a reflexive bilinear form. Show that it is either symmetric or alternat- ing. (Hint: consider g(v, g(v, w)x − g(v, x)w).)

(3) Compute the indices of inertia for the following matrices:  1 0 −1 2  −1 −1 −1  1 −2 −1 0 2 1 −2 A =   ,B = −1 0 1 ,C = −2 4 2 −1 1 0 0        −1 1 0 −1 2 −3 2 −2 0 −1

(4) Let V be a finite-dimensional K-vector space, and let g(v, w) be a non-degenerate, symmetric bilinear form. Let W ⊂ V be a subspace, then show that (W ⊥)⊥ = W .

T (5) Show that g(A, B) := Tr(A B) is a symmetric, positive definite bilinear form on Mn(R).

(6) Let g(v, w) be a (fixed) non-degenerate bilinear form on a finite-dimensional K-vector space V . (a) Show that the assignment A 7→ g(Av, w) defines a vector space isomorphism End(V ) =∼ Bilin(V × V, K).

(b) Show that for each A ∈ End(V ), there exists a unique linear transformation tA : V → V so that g(Av, w) = g(v, tAw). This is the adjoint or transpose with respect to g.

(c) Show that if g(v, w) is the dot product on Kn, then tA = AT .

 2 −1 (d) Let g(v, w) be given on R2 × R2 by the 2 × 2 matrix . Find tA for the linear −1 1 1 1 transformation given by A = . 0 1

1 2

(7) Show that any hermitian form h(v, w) on a finite-dimensional C-vector space can be written as h(v, w) = g(v, w) + if(v, w), where g, f are bilinear forms on the underlying R-vector space, g is symmetric, and f is alternating.

(8) Let g(v, w) be an inner product on a finite-dimensional R-vector space V (i.e. a symmetric, positive-definite bilinear form). We say that a linear transformation U : V → V is unitary if t −1 U = U . Show that V = V1 ⊥ · · · ⊥ Vk where each Vi is U-invariant and has dimension one or two. Further, show that if we take a g-orthonormal basis on a 2-dimensional Vi, then in this basis U is given by a matrix of the form cos θ − sin θ −1 0 cos θ − sin θ or sin θ cos θ 0 1 sin θ cos θ