Quadratic forms. Here. Thus symmetric matrices are diagonalizable, and the diagonalization can be performed by means of an orthogonal matrix.

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1 Quadratic forms 1. Symmetric matrices An n n matrix (a ij ) n ij=1 with entries on R is called symmetric if A T, that is, if a ij = a ji for all 1 i, j n. We denote by S n (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 necessarily symmetric), then for any vectors x, y R n (thought of as columnvectors), we have y Ax = y T Ax = x T A T 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 statement: Theorem 1.2. Let n 1 be an integer and let A S n (R) be a symmetric matrix. Then the following hold: (1) For any vectors x, y R n we have Ax y = x Ay. (2) The characteristic polynomial p A (t) of A completely factors out over R, that is, there exist (not necessarily distinct) λ 1,..., λ n R such that p A (t) = (t λ 1 )... (t λ r ). (3) If λ λ are distinct eigenvalues of A and x, x R n are nonzero eigenvectors corresponding to λ, λ accordingly (that is, Ax = λx, Ax = λ x ) then x x = 0. (4) If λ is an eigenvalue of A with multiplicity k 1 in p A (t), then there exist k linearly independent eigenvectors of A corresponding to λ, and the eigenspace E λ = {x R n Ax = λx} has dimension k. (5) There exists an orthonormal basis of R n consisting of eigenvectors of A. (6) Let p A (t) = (t λ 1 )... (t λ r ) be the factorization of p A (t) given by part (2) above, and let v 1,..., v r R n be an orthonormal basis of R n such that Av i = λ i v i (such a basis exists by part (5)). Let C = [v 1... v n ] be the n n matrix with columns v 1,... v n. Then C O(n) and Here C 1 AC = Diag(λ 1,..., λ n ). λ λ Diag(λ 1,..., λ n ) = λ n Thus symmetric matrices are diagonalizable, and the diagonalization can be performed by means of an orthogonal matrix. 1

2 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. and Saying that Q is bilinear means that Q(c 1 v 1 + c 2 v 2, w) = c 1 Q(v 1, w) + c 2 Q(v 2, w) Q(v, c 1 w 1 + c 2 w 2 ) = c 1 Q(v, w 1 ) + c 2 Q(v, w 2 ) for all c 1, c 2 R and v 1, v 2, v, w, w 1, w 2 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: (1) The standard dot-product : R n R n R, (x 1,..., x n ) (y 1,..., y n ) = x 1 y x n y n, is a quadratic form on R n. (2) Let A be an n n matrix with entries in R. Define Q A : R n R n R as Q A (x, y) := y T Ax = y Ax = Ay x = n a ijy i x j. Then Q A is a quadratic form on R n. (3) Let A be an n n matrix with entries in R and let B = 1 2 (A + AT ) (so that B = B T and B is symmetric). Then Q Q B, that is, for all x, y R n we have Q A (x, y) = Q B (x, y). (4) Let V be a vector space with a basis E = e 1,..., e n and let λ 1,..., λ n R be real numbers. Define Q : V V R as Q(x, y) = n i=1 λ iy i x i, where x = n i=1 y ie i are arbitrary vectors in V. Then Q is a quadratic form on V. (5) Let V be a vector space with a basis E = e 1,..., e n and let A S n (R) be a symmetric matrix. Define Q A,E : V V R as Q A,E (x, y) := a ij y i x j where x = n i=1 y ie i are arbitrary vectors in V. Then Q A,E is a quadratic form on V. Proposition 2.3. The following hold: (1) For any quadratic form Q on R n there exists a unique symmetric matrix A S n (R) such that Q = Q A.

3 (2) Let V be a vector space of dimension n with a basis E = e 1,..., e n. Let Q : V V R be a quadratic form on V. Then there exists a unique symmetric matrix (a ij ) ij S n (R) such that Q = Q A,E. Moreover a ij = Q(e i, e j ) 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 a ii = Q(e i, e i ) and for any 1 ij n, i j we have a ij = 1 2 (Q(e i + e j, e i + e j ) Q(e i, e i ) Q(e j, e 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 associated 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. Let E = e 1,..., e n and E = e 1,..., e n be two bases of V. Let A, A be symmetric matrices such that Q = Q A,E = Q A,E. Then A = C T AC, where C is the transition matrix from E to E, that is e j = n i=1 c ije j. 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 = e 1,..., e n and real numbers λ 1 λ n such that Q = Q A,E with Diag(λ 1,..., λ n ). This means that Q(x, y) = n i=1 λ ix i y i where x = n i=1 y ie i are arbitrary vectors in V. The values λ 1,..., λ n in Corollary 2.5 are not uniquely determined by Q. However, it turns out that the number n + of i {1..., n} such that λ i > 0, the number n of i {1..., n} such that λ i < 0 and the number n 0 of i {1..., n} such that λ i = 0 are uniquely determined by Q. The triple (n 0, n +, n ) is called the signature of Q. Note that by definition, n 0 + 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 0. (2) We have n + = n and n 0 = n = 0. Remark 2.7. Thus we see that if Q is a quadratic form on V with Q = Q A,E for some basis E of V and a symmetric matrix A S n (R) with eigenvalues λ 1,..., λ n R, then Q is positive-definite if and only if λ i > 0 for i = 1,..., n (in which case we also have det(a) = λ 1... λ n > 0). 3

4 4 Proposition 2.6 implies that if Q is a positive-quadratic form on a vectorspace 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 = e 1, e 2 be a basis of V and let Q : V V R be a quadratic form on V. Then Q = Q A,E where E F F G where E = Q(e 1, e 1 ), F = Q(e 1, e 2 ) = Q(e 2, e 1 ) and G = Q(e 2, e 2 ). For x = x 1 e 1 + x 2 e 2 V and y = y 1 e 1 + y 2 e 2 V we have Q(x, y) = i,j = 1 n a ij y i x j = Ex 1 y 1 + F x 1 y 2 + F x 2 y 1 + Gx 2 y 2 and Q(x, x) = Ex F x 1x 2 + Gx 2 2. Example 3.1. (a) Let Q : R 2 R 2 we given by Q((x 1, x 2 ), (y 1, y 2 )) = x 1 y 1 + 3x 1 y 2 + 3x 2 y 1 + x 2 y 2. Thus for the standard unit basis E = e 1, e 2 of R 2 we have Q = Q A,E where so that E = G = 1, F = 3.The characteristic polynomial of A is λ 2 2λ 8 = (λ 4)(λ + 2), with roots λ 1 = 2, λ 2 = 4. Thus the signature of Q is n + = n = 1, n 0 = 0, and the quadratic form Q is not positive-definite. (b) Let Q : R 2 R 2 we given by Q((x 1, x 2 ), (y 1, y 2 )) = x 1 y 1 + 2x 1 y 1 + 2x 2 y 1 + 7x 2 y 2. Thus for the standard unit basis E = e 1, e 2 of R 2 we have Q = Q A,E where so that E = G = 1, F = 3.The characteristic polynomial of A is λ 2 8λ + 3, with roots λ 1 = 4 13, λ 2 = Since λ 1 > 0, λ 2 > 0, the signature of Q is n + = 2, n 0 = n = 0, and the quadratic form Q is positive-definite Riemannian metrics. A Riemannian metric g on R n is a function that, to every p R n, associates a positive-definite quadratic form g(p) on T p R n. Thus for every p R n g p has the form g p = Q A(p) for some symmetric n n matrix A(p) = (g ij (p)) ij, so that g p ((x 1,..., x n ) p, (y 1,..., y n ) p ) = g ij (p)y i x j

5 and g p ((x 1,..., x n ) p, (x 1,..., x n ) p ) = g ij (p)x i x j. Note that g ij (p) = g p ((e i ) p, (e j ) p ). We think of g p as an inner product on T p R n. In the case n = 2 the matrix A(p) is E(p) F (p) F (p) G(p) so that g p ((x 1, x 2 ) p, (y 1, y 2 ) p ) = E(p)x 1 y 1 + F (p)x 1 y 2 + F (p)x 2 y 1 + G(p)x 2 y 2 and g p ((x 1, x 2 ) p, (x 1, x 2 ) p ) = E(p)x F (p)x 1 x 2 + G(p)x 2 2. A Riemannian metric g on R n is smooth if the functions g ij : R n R are smooth for all 1 i, j n. Let g be a smooth Riemannian metric on R n. If v p T p R n, we put v p g := g p (v p, v p ) and call this number the g-norm of v p. For a smooth curve γ : [a, b] R n, γ(t) = (x 1 (t),..., x n (t)), we define the length of γ with respect to g as b b s := γ (t) g dt = g ij (γ(t)) dx i dx j dt dt dt a a For this reason, the Riemannian metric g is often symbolically denoted as ds 2 = g ij dx i dx j. 5