Quadratic Forms

Quadratic forms

1. Symmetric matrices n An n × n matrix A = (aij)ij=1 with entries on R is called symmetric if T A = A , that is, if aij = aji for all 1 ≤ i, j ≤ n. We denote by Sn(R) the set of all n × n symmetric matrices with coeﬃcients in R. Remark 1.1. Note that if B is an n × n matrix with entries in R (not n necessarily symmetric), then for any vectors x, y ∈ R (thought of as column- vectors), we have y · Ax = yT Ax = xT AT y. Here we think of the dot-product y · Ax as a 1 × 1 matrix. We summarize the properties of symmetric matrices in the following state- ment:

Theorem 1.2. Let n ≥ 1 be an integer and let A ∈ Sn(R) be a symmetric matrix. Then the following hold: n (1) For any vectors x, y ∈ R we have Ax · y = x · Ay. (2) The characteristic polynomial pA(t) of A completely factors out over R, that is, there exist (not necessarily distinct) λ1, . . . , λn ∈ R such that pA(t) = (t − λ1) ... (t − λr). 0 0 n (3) If λ 6= λ are distinct eigenvalues of A and x, x ∈ R are nonzero eigenvectors corresponding to λ, λ0 accordingly (that is, Ax = λx, Ax0 = λ0x0) then x · x0 = 0. (4) If λ is an eigenvalue of A with multiplicity k ≥ 1 in pA(t), then there exist k linearly independent eigenvectors of A corresponding to n λ, and the eigenspace Eλ = {x ∈ R |Ax = λx} has dimension k. n (5) There exists an orthonormal basis of R consisting of eigenvectors of A. (6) Let pA(t) = (t − λ1) ... (t − λr) be the factorization of pA(t) given n by part (2) above, and let v1, . . . , vr ∈ R be an orthonormal basis n of R such that Avi = λivi (such a basis exists by part (5)). Let C = [v1| ... |vn] be the n × n matrix with columns v1, . . . vn. Then C ∈ O(n) and −1 C AC = Diag(λ1, . . . , λn). Here   λ1 0 0 ... 0  0 λ2 0 ... 0    Diag(λ1, . . . , λn) =  ......   . . . . .  0 0 0 . . . λn Thus symmetric matrices are diagonalizable, and the diagonalization can be performed by means of an orthogonal matrix. 1 2

2. Quadratic forms Deﬁnition 2.1 (Quadratic form). Let V be a vector space over R of ﬁnite dimension n ≥ 1. A quadratic form on V is a bilinear map Q : V × V → R such that Q is symmetric, that is, Q(v, w) = Q(w, v) for all v, w ∈ V . Saying that Q is bilinear means that

Q(c1v1 + c2v2, w) = c1Q(v1, w) + c2Q(v2, w) and Q(v, c1w1 + c2w2) = c1Q(v, w1) + c2Q(v, w2) for all c1, c2 ∈ R and v1, v2, v, w, w1, w2 ∈ V . To every quadratic form Q : V ×V → R we also associate the “quadratic” function q : V → R defined as q(v) := Q(v.v) for v ∈ V . We will see in Proposotion 2.3 below that the quadratic form Q can be uniquely recovered from q. Example 2.2. We have: n n (1) The standard dot-product · : R ×R → R,(x1, . . . , xn)·(y1, . . . , yn) = n x1y1 + ··· + xnyn, is a quadratic form on R . (2) Let A be an n × n matrix with entries in R. Define n n QA : R × R → R T Pn as QA(x, y) := y Ax = y · Ax = Ay · x = i,j=1 aijyixj. n Then QA is a quadratic form on R . 1 T (3) Let A be an n × n matrix with entries in R and let B = 2 (A + A ) T (so that B = B and B is symmetric). Then QA = QB, that is, for n all x, y ∈ R we have QA(x, y) = QB(x, y). (4) Let V be a vector space with a basis E = e1, . . . , en and let λ1, . . . , λn ∈ R be real numbers. Pn Define Q : V × V → R as Q(x, y) = i=1 λiyixi, where x = Pn Pn i=1 xiei and y = i=1 yiei are arbitrary vectors in V . Then Q is a quadratic form on V . (5) Let V be a vector space with a basis E = e1, . . . , en and let A ∈ Sn(R) be a symmetric matrix. Define QA,E : V × V → R as n X QA,E (x, y) := aijyixj i,j=1 Pn Pn where x = i=1 xiei and y = i=1 yiei are arbitrary vectors in V . Then QA,E is a quadratic form on V . Proposition 2.3. The following hold: n (1) For any quadratic form Q on R there exists a unique symmetric matrix A ∈ Sn(R) such that Q = QA. 3

(2) Let V be a vector space of dimension n with a basis E = e1, . . . , en. Let Q : V × V → R be a quadratic form on V . Then there exists a unique symmetric matrix A = (aij)ij ∈ Sn(R) such that Q = QA,E . Moreover aij = Q(ei, ej) for all 1 ≤ i, j ≤ n. (3) Let V, Q, E and A be as in (2) above. Then for i = 1, . . . , n we have aii = Q(ei, ei) and for any 1 ≤ ij ≤ n, i 6= j we have 1 a = (Q(e + e , e + e ) − Q(e , e ) − Q(e , e )). ij 2 i j i j i i j j Thus part (3) of Proposition 2.3 shows that if Q : V × V is a quadratic form, then we can recover Q from knowing the values Q(x, x) for all x ∈ V . Proposition 2.3 describes all quadratic forms on V in terms of their as- sociated symmetric matrices, once we choose a basis E of V . It is useful to know how these matrices change when we change the basis of V . Proposition 2.4. Let V be a vector space of dimension n and let Q : V × V → R be a quadratic form on V . 0 0 0 0 Let E = e1, . . . , en and E = e1, . . . , en be two bases of V . Let A, A be symmetric matrices such that Q = QA,E = QA0,E0 . Then A0 = CT AC, 0 0 Pn where C is the transition matrix from E to E , that is ej = i=1 cijej. Together with Theorem 1.2, Proposition 2.4 implies: Corollary 2.5. Let V be a vector space of dimension n and let Q : V ×V → R be a quadratic form on V . Then there exist a basis E = e1, . . . , en and real numbers λ1 ≤ · · · ≤ λn such that Q = QA,E with A = Diag(λ1, . . . , λn). This Pn Pn Pn means that Q(x, y) = i=1 λixiyi where x = i=1 xiei and y = i=1 yiei are arbitrary vectors in V .

The values λ1, . . . , λn in Corollary 2.5 are not uniquely determined by Q. However, it turns out that the number n+ of i ∈ {1 . . . , n} such that λi > 0, the number n− of i ∈ {1 . . . , n} such that λi < 0 and the number n0 of i ∈ {1 . . . , n} such that λi = 0 are uniquely determined by Q. The triple (n0, n+, n−) is called the signature of Q. Note that by definition, n0 + n+ + n− = n. Proposition 2.6. Let V be a vector space over R of finite dimension n ≥ 1. Then the following are equivalent: (1) The form Q is positive-definite, that is, Q(x, x) > 0 for every x ∈ V, x 6= 0. (2) We have n+ = n and n0 = n− = 0.

Remark 2.7. Thus we see that if Q is a quadratic form on V with Q = QA,E for some basis E of V and a symmetric matrix A ∈ Sn(R) with eigenvalues λ1, . . . , λn ∈ R, then Q is positive-deﬁnite if and only if λi > 0 for i = 1, . . . , n (in which case we also have det(A) = λ1 . . . λn > 0). 4

Proposition 2.6 implies that if Q is a positive-quadratic form on a vector- space V of ﬁnite dimension n ≥ 1, then for every nonzero linear subspace W of V the restriction of Q to W , Q : W × W → R, is again a positive-deﬁnite quadratic form.

3. The case of dimension 2

Let V be a vector space over R of dimension 2. Let E = e1, e2 be a basis of V and let Q : V × V → R be a quadratic form on V . Then Q = QA,E where EF A = FG where E = Q(e1, e1), F = Q(e1, e2) = Q(e2, e1) and G = Q(e2, e2). For x = x1e1 + x2e2 ∈ V and y = y1e1 + y2e2 ∈ V we have

X n Q(x, y) = = 1 aijyixj = Ex1y1 + F x1y2 + F x2y1 + Gx2y2 i,j 2 2 and Q(x, x) = Ex1 + 2F x1x2 + Gx2. 2 2 Example 3.1. (a) Let Q : R → R we given by Q((x1, x2), (y1, y2)) = x1y1 + 3x1y2 + 3x2y1 + x2y2. Thus for the standard unit basis E = e1, e2 of R2 we have Q = QA,E where 1 3 A = 3 1 so that E = G = 1, F = 3.The characteristic polynomial of A is λ2−2λ−8 = (λ − 4)(λ + 2), with roots λ1 = −2, λ2 = 4. Thus the signature of Q is n+ = n− = 1, n0 = 0, and the quadratic form Q is not positive-definite. 2 2 (b) Let Q : R → R we given by Q((x1, x2), (y1, y2)) = x1y1 + 2x1y1 + 2x2y1 + 7x2y2. Thus for the standard unit basis E = e1, e2 of R2 we have Q = QA,E where 1 2 A = 2 7 2 so that E = G = 1, F√= 3.The characteristic√ polynomial of A is λ − 8λ + 3, with roots λ1 = 4 − 13, λ2 = 4 + 13. Since λ1 > 0, λ2 > 0, the signature of Q is n+ = 2, n0 = n− = 0, and the quadratic form Q is positive-definite. n 3.1. Riemannian metrics. A Riemannian metric g on R is a function n that, to every p ∈ R , associates a positive-definite quadratic form g(p) on n TpR . n Thus for every p ∈ R g|p has the form g|p = QA(p) for some symmetric n × n matrix A(p) = (gij(p))ij, so that n X gp((x1, . . . , xn)p, (y1, . . . , yn)p) = gij(p)yixj i,j=1 5 and n X gp((x1, . . . , xn)p, (x1, . . . , xn)p) = gij(p)xixj. i,j=1 Note that gij(p) = g|p((ei)p, (ej)p). We think of g|p as an “inner product” n on TpR . In the case n = 2 the matrix A(p) is E(p) F (p) A = F (p) G(p) so that

g|p((x1, x2)p, (y1, y2)p) = E(p)x1y1 + F (p)x1y2 + F (p)x2y1 + G(p)x2y2 and 2 2 g|p((x1, x2)p, (x1, x2)p) = E(p)x1 + 2F (p)x1x2 + G(p)x2. n n A Riemannian metric g on R is smooth if the functions gij : R → R are smooth for all 1 ≤ i, j ≤ n. n n Let g be a smooth Riemannian metric on R . If vp ∈ TpR , we put p ||vp||g := g|p(vp, vp) and call this number the g-norm of vp. n For a smooth curve γ :[a, b] → R , γ(t) = (x1(t), . . . , xn(t)), we deﬁne the length of γ with respect to g as v Z b Z b u n X dxi dxj s := ||γ0(t)|| dt = u g (γ(t)) dt g t ij dt dt a a i,j=1 For this reason, the Riemannian metric g is often symbolically denoted as n 2 X ds = gijdxidxj. i,j=1