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1 Bilinear Forms Bilinear forms and quadratic forms. Definition 1.3. A symmetric bilinear form on the vector space A Review of Linear Algebra. V is a map q : V V R satisfying the following: × → Definition 1.1. A square matrix P of real components For each fixed x V , both is said to be an• orthogonal matrix if tPP = P tP = I • ∈ t q(x, ): V y q(x, y) R and holds, where P denotes the transposition of P and I is · ∋ 7→ ∈ the identity matrix. q( , x): V y q(y, x) R · ∋ 7→ ∈ A square matrix A is said to be (real) symmetric matrix are linear maps. • t if A = A holds. For any x and y V , q(x, y) = q(y, x) holds. • ∈ Fact 1.2. The eigenvalues of a real symmetric matrix are The quadratic form associated to the symmetric bilinear form q • real numbers, and the dimension of the corresponding eigenspace is a mapq ˜: V x q(x, x) R. ∋ 7→ ∈ coincides with the multiplicity of the eigenvalue. Lemma 1.4. A quadratic form determines the symmetric bi- linear form. In other words, two symmetric bilinear forms with Real symmetric matrices can be diagonalized by orthogonal common quadratic form coincide with each other. • matrices. In other words, for each real symmetric matrix A, there exists an orthogonal matrix P satisfying Proof. Let q be a symmetric bilinear form andq ˜ the quadratic form associated to it. Since 1 t P − AP = P AP = diag(λ1, . . . , λn), q˜(x + y) = q(x + y, x + y) where diag(... ) denotes the diagonal matrix with diago- = q(x, x) + q(x, y) + q(y, x) + q(y, y) nal components “... ”. In particular, λ1, . . . , λn are the =q ˜(x) + 2q(x, y) +q ˜(y) eigenvalues of A counted with their multiplicity.{ } holds for each x, y V , we have ∈ In this section, V denotes an n-dimensional vector space over 1 R (n < ). q(x, y) = q˜(x + y) q˜(x) q˜(y) . ∞ 2 − − 09. April, 2019. Revised: 16. April, 2019 Hence the symmetric bilinear( form q is determined) byq ˜. 3 (20190721) MTH.B405; Sect. 1 MTH.B405; Sect. 1 (20190721) 4

By virtue of Lemma 1.4, a symmetric bilinear form itself is Lemma 1.6. For a symmetric bilinear form q on V , there exists often called a quadratic form. the unique n n symmetric matrix A satisfying ×

Example 1.5. For an n n symmetric matrix A = (aij) with t × q(x, y) = qA(xˆ, yˆ) = xˆAyˆ, real components and column vectors xˆ, yˆ Rn, we set ∈ ( ) n n t n (1.1) qA : R R (xˆ, yˆ) xˆAyˆ R, where xˆ and yˆ R are the components of x, y with respect to × ∋ 7−→ ∈ ∈ [v1,..., vn], respectively where txˆ the column vector obtained by transposing xˆ. Then n Proof. Setting A = (aij) by aij := q(vi, vj), the conclusion qA is a symmetric bilinear form on R . In particular, qI is the n follows. In fact, canonical inner product of R , where I is the identity matrix. Conversely, for each symmetric bilinear form q in n, there R n n n exists a symmetric matrix A such that q = qA. In fact, setting q(x, y) = q x v , y v = x y q(v , v ) a := q(e , e ), A = (a ) satisfies q = q , where [e ] is the  i i j j i j i j ij i j ij A j i=1 j=1 i,j=1 canonical basis of n. ∑ ∑ ∑ R n  t = xiyjaij = xˆAyˆ = qA(xˆ, yˆ), Matrix representation of quadratic forms. Take a basis i,j=1 ∑ [v1,..., vn] of the vector space V . Then t t where xˆ = (x1, . . . , xn) and yˆ = (y1, . . . , yn). x 1 We call the matrix A in Lemma 1.6 the representative matrix . n (1.2) V x xˆ = . R , ∋ 7−→  .  ∈ of q with respect to the basis [vj]. xn   Lemma 1.7. Take two bases [v1,..., vn], [w1,..., wn] of V   x ij 1 and let U = (u ) GL(n, R) be the basis change matrix: x = x v + + x v = [v ,..., v ] . ∈  1 1 ··· n n 1 n  .  x [w1,..., wn] = [v1,..., vn]U   n    n n gives an isomorphism of vector spaces. We call xˆ R the i.e., wj = uijvi (j = 1, . . . , n), ∈ i=1 component of x with respect to the basis [vj]. ∑ 5 (20190721) MTH.B405; Sect. 1 MTH.B405; Sect. 1 (20190721) 6 where GL(n, R) denotes the general linear group, that is, the set Remark 1.10. A positive (resp. negative) quadratic form is non- of n n regular matrix of real components. Let A (resp. A˜) be degenerate. In fact, if q(x, y) = 0 holds for all y, q(x, x) = 0 the representative× matrix of a symmetric bilinear form q with holds. On the other hand, q(x, x) > 0 (resp. < 0) when x = 0. ̸ respect to the basis [vj](resp. [wj]). Then it holds that Hence x = 0. ˜ t A = UAU. Signature of non-degenerate quadratic forms. Proof. Writing x and y as Proposition 1.11. A quadratic form q on V is positive definite (resp. positive semi-definite, negative definite, non-degenerate) x = [v1,..., vn]xˆ = [w1,..., wn]x˜, if and only if all eigenvalues of the representative matrix of q are y = [v1,..., vn]yˆ = [w1,..., wn]y˜, positive (resp. non-negative, negative, non-zero). This condition does not depend on choice of bases. we have Proof. Let [v ,..., v ] be a basis of V and A the representative xˆ = Ux˜, yˆ = Uy˜. 1 n matrix of q with respect to it. Since A is a symmetric matrix, t Hence q(x, y) = xˆAyˆ = tx˜tUAUy˜. there exists an orthogonal matrix P such that

Non-degenerate quadratic forms. λ1 ... 0 t . . . P AP = Λ, Λ := . .. . , Definition 1.8. A symmetric bilinear form (a quadratic form)  . .  0 . . . λ q on V is said to be  n  tPP =I = the identity matrix, positive definite (resp. positive semi definite) if q(x, x) > 0 • (resp. 0) holds for all x V 0 , where λ1,. . . , λn are eigenvalues of A, which are real numbers. ≧ ∈ \{ } Then, by setting [w1,..., wn] := [v1,..., vn]P , the representa- negative definite if q is positive definite, • − tive matrix of q with respect to [wj] is the diagonal matrix Λ. Denoting the components of vectors x,y with respect to [wj] by non-degenerate when “q(x, y) = 0 for all y V ” implies t t • “x = 0”. ∈ xˆ = (x1, . . . , xn) and yˆ = (y1, . . . , yn), respectively, n Example 1.9. An inner product (in the undergraduate Linear q(x, y) = λjxjyj. Algebra course) is nothing but a positive definite quadratic form. j=1 ∑ 7 (20190721) MTH.B405; Sect. 1 MTH.B405; Sect. 1 (20190721) 8

The conclusion is obtained by this equality. In fact, if q is pos- The pair (n+, n ) is called the signature of q. itive (resp. negative) definite, q(w , w ) = λ is positive (resp. − j j j Example 1.13. A positive (resp. negative) definite quadratic negative) for each j = 1, . . . , n. Hence all eigenvalues of A are form q on V has signature (n, 0) (resp. (0, n)), where n = dim V . positive (resp. negative). Conversely, if all eigenvalues are posi- tive (resp. negative), Theorem 1.14. Let (n+, n ) be the signature of a non-degenerate − n quadratic form q on V . Then n+ (resp. n ) is the number of − 2 q(x, x) = λj(xj) positive (resp. negative) eigenvalues of the representative matrix j=1 of q. In particular, n+ + n = n = dim V holds. ∑ − is positive (resp. negative). The conclusion for positive semi- Proof. As seen in the proof of Proposition 1.11 we may assume definite case is obtained in the same way. that the matrix representative with respect to the basis [wj] is a On the other hand, let q be a non-degenerate quadratic form diagonal matrix Λ, without loss of generality. Since all diagonal and assume λj = 0 for some j = 1, . . . , n. Then components of Λ are non-zero, we may assume that λ1, . . . , λt are negative, and λt+1, . . . , λn are positive. Then q is negative q(wj, wj) = λj = 0, ̸ definite on the subspace generated by w1,..., wt , and hence { } contradiction to non-degeneracy. Conversely, assume λj = 0 n ≧ t. On the other hand, q is positive definite on the subspace ̸ − (j = 1, . . . , n), and q(x, y) = 0 for all y V . Then generated by w ,..., w , and then n n t: ∈ { t+1 n} + ≧ − 0 = q(x, wj) = λjxj (1.3) n ≧ t, n+ ≧ n t. − − holds, which implies x = 0, for j = 1, . . . , n. Thus, x = 0.b j Here, by definition, there exists a subspace W+ (resp. W ) Hence q is non-degenerate. − of V such that q W+ (resp. q W ) is positive (resp. negative) | | − definite and dim W+ = n+ (resp. dim W = n ). Take a vector Let W be a linear subspace of the vector space V . Then a − − x W+ W . Then q(x, x) ≦ 0 and q(x, x) ≧ 0 hold, that is, symmetric bilinear form q : V V R on V induces a symmet- ∈ ∩ − × → q(x, x) = 0. Noticing q W is positive definite, x = 0. Hence ric bilinear form q W : W W R on W . | + | × → W+ W = 0 , and then we have Definition 1.12. For a non-degenerate quadratic form q on V , ∩ − { } we define n+ + n = dim W+ + dim W ≦ dim V = n. − − n := max dim W ; q is positive definite , Therefore (1.3) yields + { |W } n := max dim W ; q W is negative definite . − n ≧ t, n n ≧ n t n n+ ≧ t, n+ ≧ n t, { | } − − − − − − 9 (20190721) MTH.B405; Sect. 1 MTH.B405; Sect. 1 (20190721) 10

that is, n = t, n+ = n t. expressed as − − t Is O (1.5) v, w s,t = vJs,tw Js,t := − . Remark 1.15. By Theorem 1.14, the number of positive (resp. ⟨ ⟩ OIt negative) eigenvalues of the matrix representative does not de- ( ) n n pend on choice of bases. This fact is equivalent to “the number In particular, the case of signature (n, 0), R := R0 is called of positive (negative) eigenvalues of a symmetric matrix A is the Euclidean vector space, and when the signature is (n 1, 1), t n − invariant to the transformation A UAU (U is a regular ma- the space R1 is called the Minkowski vector space. trix)”. 7→ Orthonormal basis In this paragraph, we fix an inner prod- Definition 1.16. An inner product on a finite dimensional vec- uct , on V . ⟨ ⟩ tor space V is a non-degenerate quadratic (symmetric bilinear) Definition 1.17. A vector v V is said to be orthogonal to form. A vector space with fixed inner product is called an inner w V if v, w = 0 holds. ∈ product space or a metric space. ∈ ⟨ ⟩ Definition 1.18. An n-tuple e1,..., en of V is called an orthonormal basis of V if { } Pseudo Euclidean vector spaces. Let s ≧ 0, t ≧ 0 be e , e = δ (1 i, j n) integers satisfying n := s + t ≧ 2. Then a quadratic form | ⟨ i j⟩ | ij ≦ ≦ holds. s s+t Lemma 1.19. An orthonormal basis is a basis of V . (1.4) v, w := v w + v w ⟨ ⟩s,t −  j j  k k j=1 k=s+1 Proof. It is sufficient to show linear independency. ∑ ∑     v1 w1 Theorem 1.20. There exists an orthonormal basis for an arbi- . . v =  .  , w =  .  trary inner product space. In particular, if the signature of the v w inner product is (t, s), one can take a basis [ej] satisfying   n  n      ei, ej = 0 (i = j), gives an inner product on Rn with signature (t, s). We denote by ⟨ ⟩ ̸ n 1 (i = 1, . . . , s) Rs such an inner product space, and call the pseudo Euclidean ei, ei = − vector space of signature (s, t). The inner product (1.4) can be ⟨ ⟩ { 1 (i = s + 1, . . . , s + t). 11 (20190721) MTH.B405; Sect. 1 MTH.B405; Sect. 1 (20190721) 12

Proof. As seen in the proof of Proposition 1.11, there exists a Exercises basis [wj] such that the matrix representative of , is an diag- ⟨ ⟩ 1-1 For an m n matrix C, we set A := tCC, which is an onal matrix Λ = diag(λ1, . . . , λn). Since the number of positive n n-symmetric× matrix. Let q be the quadratic form on (resp. negative) eigenvalues is t (resp. s), we may assume × A Rn as in Example 1.5 induced from A. < 0 (j = 1, . . . , s) (1) Prove that q is positive semi-definite. λj A > 0 (j = s + 1, . . . , n) { (2) Find a condition of C for qA to be positive definite. without loss of generality. We set 1-2 Let M2(R) be the set of 2 2 real matrices, and × t U := diag 1/ λ1 ,..., 1/ λn . Sym(2, R) := A M2(R); A = A , | | | | { ∈ } ( √ √ ) Sym+(2, R) := A Sym(2, R); qA is positive definite , Then it holds that { ∈ } where qA is the quadratic form as defined in Example 1.5. t Is O (1.6) UΛU = , Is the subset Sym+(2, R) a smooth submanifold of M2(R) = −OI ( t) R4? where Im is the m m identity matrix, and O denotes the zero matrix of an appropriate× size. Hence by Lemma 1.7, the matrix representative of , with respect to the basis [e ,..., e ] := ⟨ ⟩ 1 n [w1,..., wn]U is the matrix in (1.6). Hence [ej] satisfies the desired property.