Quadratic forms. Defn

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1 Quadratic forms Aim lecture: We use the spectral thm for self-adjoint operators to study solns to some multi-variable quadratic eqns. In this lecture, we work over the real field F = R. We use the notn x = x 1,..., x n ), y = y 1,..., y n ) R n Defn A real) quadratic form is a function Q : R n R of the form Qx) = i j β ijx i x j for some β ij R. A quadric in R n is a non-empty set of the form V Q) = {x R n Qx) = 1} for some given quadratic form Q i.e. V Q) is the set of solns to Qx) = 1. Rem If Q is a quadratic form & d R 0 such that the solns V to Qx) = d is non-empty, then V is the quadric V 1 d Q). E.g. For a ij R, consider the function Q x 1 ) x 2 ) = a11 a12 x1 x 2 ) a 21 a 22 ) x 1 ) x 2 = Daniel Chan UNSW) Lecture 40: Quadrics Semester / 10

2 Quadratic forms via symmetric matrices Quadratic forms can be studied via matrices because of the following Prop-Defn 1 Let A M nn R). Then Q A : R n R : x x T Ax is a quadratic form. 2 Let Q : R n R be a quadratic form. Then there is a unique symmetric matrix A M nn R) such that Q = Q A. In fact if A = a ij ) ij then Q A x) = i a ii x 2 i + a ij + a ji )x i x j. i<j Proof. Just generalise the calculation in the previous example. E.g. Daniel Chan UNSW) Lecture 40: Quadrics Semester / 10

3 Non-degenerate forms Recall that the e-values of a real symmetric matrix are all real even when considered as a complex matrix). Defn Let Q : R n R be a quadratic form & A M nn R) be the symmetric matrix with Q = Q A. Let τ +, τ be the number of positive resp negative e-values of A, counted with multiplicity alg = geom). We say Q or V Q)) is non-degenerate if A is invertible. Otherwise we say it is degenerate. The rank of Q is rankq = rank A = τ + + τ. The signature of Q is τ + τ. E.g. A degenerate quadric in R 2 looks like Daniel Chan UNSW) Lecture 40: Quadrics Semester / 10

4 Conics Defn A non-degenerate quadric in R 2 is called a conic. Hopefully, you already know the following examples. Below we let a 1, a 2 > 0. E.g. 1 If D = 1) 1) = I 2 then V Q D ) is the circle x1 2 + x 2 2 = 1 & Q D has signature 2. E.g. 2 If D = 1 ) 1 ) then V Q a1 2 a2 2 D ) is the ellipse x1 + x2 = 1 & Q D has signature 2. a 1 a 2 N.B. For a 1 a 2, the x 1 and x 2 -intercepts give the points closest and furthest from 0, 0). Daniel Chan UNSW) Lecture 40: Quadrics Semester / 10

5 Conics cont d E.g. 3 If D = 1 ) 1 ) then V Q a1 2 a2 2 D ) is the hyperbola x1 x2 = 1 a 1 a 2 & Q D has signature 0. Note the following geometric features: Closest points to 0,0): Asymptotes: Daniel Chan UNSW) Lecture 40: Quadrics Semester / 10

6 Quadric surfaces A quadric surface is a non-degenerate quadric in R 3. Let a 1, a 2, a 3 > 0. Prop If D = 1 ) 1 ) 1 ) then V Q a1 2 a2 2 a3 2 D ) is the ellipsoid x1 + x2 + x3 = 1. a 1 a 2 a 3 The points closest resp furthest) from 0, 0) found by setting x i = 0 for all non-minimal resp non-maximal) a i. The signature is 3. Prop If D = 1 ) 1 ) 1 ) then V Q a1 2 a2 2 a3 2 D ) is the hyperboloid of 1-sheet x1 + x2 x3 = 1. a 1 a 2 a 3 The pts closest to 0, 0) must lie on the ellipse x 3 = 0. The signature = 1. Daniel Chan UNSW) Lecture 40: Quadrics Semester / 10

7 Quadric surfaces cont d A quadric surface is a non-degenerate quadric in R 3. Let a 1, a 2, a 3 > 0. Prop If D = 1 ) 1 ) 1 ) then V Q a1 2 a2 2 a3 2 D ) is the hyperboloid of 2-sheets x1 x2 x3 = 1. a 1 a 2 a 3 The points closest to 0, 0) are the x 1 -intercepts. The signature is -1. Sketch Let s re-scale axes by changing variables to y i = x i a i. Then in R 3 y -space the surface is just the surface of revolution of Daniel Chan UNSW) Lecture 40: Quadrics Semester / 10

8 Principal axis theorem Let A M nn R) be a symm matrix. We apply the spectral thm for self-adjoint operators to orthog diagonalise D = U T AU where U O n & D = i λ i ). Let v i be the i-th column of U, a unit norm e-vector of A with e-value λ i. Consider now the orthonormal change of co-ord system U : R n y R n x & introduce new co-ords y = U 1 x = U T x. In our new co-ords, our quadratic form Q A becomes Q A x) = x T Ax = Uy) T UDU T )Uy) = y T U T UDU T Uy = y T Dy = Q D y). We obtain Theorem Principal axis) There is an orthonormal change of co-ords such that the quadric V Q A ) has the form V Q D ) for D the diagonal matrix of e-values. The new co-ord axes called principal axes go through the corresponding e-vectors v i. In particular, any conic is either an ellipse or an hyperbola depending on whether the signature of A is 2 or 0. Any quadric surface is either an ellipsoid, an hyperboloid of 1-sheet, or an hyperboloid of 2-sheets, depending on whether the signature is 3,1 or -1. Daniel Chan UNSW) Lecture 40: Quadrics Semester / 10

9 Example E.g. Describe the quadric 2x x x x 1 x 2 + 4x 1 x 3 + 8x 2 x 3 = 3 Find the closest point on the quadric to the 0, 0, 0) T. A We may re-write the quadric as x T Ax = 3 where A = We orthogonally diagonalised this last lecture to see A = UDU T where D = 6) 3) 3) & U O 3. Also the e-spaces were E 6 = Span1, 2, 2) T, E 3 = E6. Hence there s an orthonormal change of co-ords U : R 3 y R 3 x so that Daniel Chan UNSW) Lecture 40: Quadrics Semester / 10

10 Example cont d Daniel Chan UNSW) Lecture 40: Quadrics Semester / 10