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Chapter 7: Identifying Second Degree Equations

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Chapter 7: Identifying Second Degree Equations Chapter 7 Identifying second degree equations 7.1 The eigenvalue method In this section we apply eigenvalue methods to determine the geometrical nature of the second degree equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, (7.1) where not all of a, h, b are zero. a h Let A = be the matrix of the quadratic form ax2 + 2hxy + by2. h b · ¸ We saw in section 6.1, equation 6.2 that A has real eigenvalues λ1 and λ2, given by a + b (a b)2 + 4h2 a + b + (a b)2 + 4h2 λ1 = − − , λ2 = − . p 2 p 2 We show that it is always possible to rotate the x, y axes to x1, y1 axes whose positive directions are determined by eigenvectors X1 and X2 corresponding to λ1and λ2 in such a way that relative to the x1, y1 axes, equation 7.1 takes the form 2 2 a′x + b′y + 2g′x + 2f ′y + c = 0. (7.2) Then by completing the square and suitably translating the x1, y1 axes, to new x2, y2 axes, equation 7.2 can be reduced to one of several standard forms, each of which is easy to sketch. We need some preliminary definitions. 129 130 CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS DEFINITION 7.1.1 (Orthogonal matrix) An n n real matrix P is × called orthogonal if t P P = In. It follows that if P is orthogonal, then det P = 1. For ± det (P tP ) = det P t det P = ( det P )2, so (det P )2 = det I = 1. Hence det P = 1. n ± If P is an orthogonal matrix with det P = 1, then P is called a proper orthogonal matrix. THEOREM 7.1.1 If P is a 2 2 orthogonal matrix with det P = 1, then × cos θ sin θ P = − sin θ cos θ · ¸ for some θ. REMARK 7.1.1 Hence, by the discusssion at the beginning of Chapter 6, if P is a proper orthogonal matrix, the coordinate transformation x x1 = P y y1 · ¸ · ¸ represents a rotation of the axes, with new x1 and y1 axes given by the repective columns of P . t Proof. Suppose that P P = I2, where ∆ = det P = 1. Let a b P = . c d · ¸ Then the equation t 1 1 P = P − = adj P ∆ gives a c d b = − b d c a · ¸ · − ¸ Hence a = d, b = c and so − a c P = − , c a · ¸ where a2 + c2 = 1. But then the point (a, c) lies on the unit circle, so a = cos θ and c = sin θ, where θ is uniquely determined up to multiples of 2π. 7.1. THE EIGENVALUE METHOD 131 a c DEFINITION 7.1.2 (Dot product). If X = and Y = , then b d X Y , the dot product of X and Y , is defined by· ¸ · ¸ · X Y = ac + bd. · The dot product has the following properties: (i) X (Y + Z)= X Y + X Z; · · · (ii) X Y = Y X; · · (iii) (tX) Y = t(X Y ); · · a (iv) X X = a2 + b2 if X = ; · b · ¸ (v) X Y = XtY . · The length of X is defined by X = a2 + b2 = (X X)1/2. || || · p We see that X is the distance between the origin O = (0, 0) and the point || || (a, b). THEOREM 7.1.2 (Geometrical meaning of the dot product) Let A = (x1, y1) and B = (x2, y2) be points, each distinct from the origin x1 x2 O = (0, 0). Then if X = and Y = , we have y1 y2 · ¸ · ¸ X Y = OA OB cos θ, · · where θ is the angle between the rays OA and OB. Proof. By the cosine law applied to triangle OAB, we have AB2 = OA2 + OB2 2OA OB cos θ. (7.3) − · 2 2 2 2 2 2 2 2 2 Now AB = (x2 x1) + (y2 y1) , OA = x + y , OB = x + y . − − 1 1 2 2 Substituting in equation 7.3 then gives 2 2 2 2 2 2 (x2 x1) + (y2 y1) = (x1 + y1) + (x2 + y2) 2OA OB cos θ, − − − · 132 CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS which simplifies to give OA OB cos θ = x1x2 + y1y2 = X Y. · · It follows from theorem 7.1.2 that if A = (x1, y1) and B = (x2, y2) are x1 x2 points distinct from O = (0, 0) and X = and Y = , then y1 y2 X Y = 0 means that the rays OA and OB·are perpendicular.¸ · This¸ is the · reason for the following definition: DEFINITION 7.1.3 (Orthogonal vectors) Vectors X and Y are called orthogonal if X Y = 0. · There is also a connection with orthogonal matrices: THEOREM 7.1.3 Let P bea2 2 real matrix. Then P is an orthogonal × matrix if and only if the columns of P are orthogonal and have unit length. t Proof. P is orthogonal if and only if P P = I2. Now if P = [X1 X2], the | matrix P tP is an important matrix called the Gram matrix of the column vectors X1 and X2. It is easy to prove that t X1 X1 X1 X2 P P =[Xi Xj]= · · . · X2 X1 X2 X2 · · · ¸ t Hence the equation P P = I2 is equivalent to X1 X1 X1 X2 1 0 · · = , X2 X1 X2 X2 0 1 · · · ¸ · ¸ or, equating corresponding elements of both sides: X1 X1 = 1, X1 X2 = 0, X2 X2 = 1, · · · which says that the columns of P are orthogonal and of unit length. The next theorem describes a fundamental property of real symmetric matrices and the proof generalizes to symmetric matrices of any size. THEOREM 7.1.4 If X1 and X2 are eigenvectors corresponding to distinct eigenvalues λ1 and λ2 of a real symmetric matrix A, then X1 and X2 are orthogonal vectors. 7.1. THE EIGENVALUE METHOD 133 Proof. Suppose AX1 = λ1X1, AX2 = λ2X2, (7.4) t where X1 and X2 are non–zero column vectors, A = A and λ1 = λ2. t 6 We have to prove that X1X2 = 0. From equation 7.4, t t X2AX1 = λ1X2X1 (7.5) and t t X1AX2 = λ2X1X2. (7.6) From equation 7.5, taking transposes, t t t t (X2AX1) = (λ1X2X1) so t t t X1A X2 = λ1X1X2. Hence t t X1AX2 = λ1X1X2. (7.7) Finally, subtracting equation 7.6 from equation 7.7, we have t (λ1 λ2)X1X2 = 0 − and hence, since λ1 = λ2, 6 t X1X2 = 0. THEOREM 7.1.5 Let A be a real 2 2 symmetric matrix with distinct × eigenvalues λ1 and λ2. Then a proper orthogonal 2 2 matrix P exists such × that t P AP = diag (λ1, λ2). Also the rotation of axes x x1 = P y y1 · ¸ · ¸ “diagonalizes” the quadratic form corresponding to A: t 2 2 X AX = λ1x1 + λ2y1. 134 CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS Proof. Let X1 and X2 be eigenvectors corresponding to λ1 and λ2. Then by theorem 7.1.4, X1 and X2 are orthogonal. By dividing X1 and X2 by their lengths (i.e. normalizing X1 and X2) if necessary, we can assume that X1 and X2 have unit length. Then by theorem 7.1.1, P = [X1 X2] is an | orthogonal matrix. By replacing X1 by X1, if necessary, we can assume − that det P = 1. Then by theorem 6.2.1, we have t 1 λ1 0 P AP = P − AP = . 0 λ2 · ¸ Also under the rotation X = PY , t t t t t X AX = (PY ) A(PY )= Y (P AP )Y = Y diag (λ1, λ2)Y 2 2 = λ1x1 + λ2y1. EXAMPLE 7.1.1 Let A be the symmetric matrix 12 6 A = − . 6 7 · − ¸ Find a proper orthogonal matrix P such that P tAP is diagonal. Solution. The characteristic equation of A is λ2 19λ + 48 = 0, or − (λ 16)(λ 3) = 0. − − Hence A has distinct eigenvalues λ1 = 16 and λ2 = 3. We find corresponding eigenvectors 3 2 X1 = − and X2 = . 2 3 · ¸ · ¸ Now X1 = X2 = √13. So we take || || || || 1 3 1 2 X1 = − and X2 = . √13 2 √13 3 · ¸ · ¸ Then if P =[X1 X2], the proof of theorem 7.1.5 shows that | 16 0 P tAP = . 0 3 · ¸ However det P = 1, so replacing X1 by X1 will give det P = 1. − − 7.1. THE EIGENVALUE METHOD 135 y y 2 4 2 x -4 -2 2 4 x -2 2 -4 Figure 7.1: 12x2 12xy + 7y2 + 60x 38y + 31 = 0. − − REMARK 7.1.2 (A shortcut) Once we have determined one eigenvec- a b tor X1 = , the other can be taken to be − , as these vectors are b a · ¸ · ¸ 2 2 always orthogonal. Also P =[X1 X2] will have det P = a + b > 0. | We now apply the above ideas to determine the geometric nature of second degree equations in x and y. EXAMPLE 7.1.2 Sketch the curve determined by the equation 12x2 12xy + 7y2 + 60x 38y + 31 = 0. − − Solution. With P taken to be the proper orthogonal matrix defined in the previous example by 3/√13 2/√13 P = , 2/√13 3/√13 · − ¸ then as theorem 7.1.1 predicts, P is a rotation matrix and the transformation x x1 X = = PY = P y y1 · ¸ · ¸ 136 CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS or more explicitly 3x1 + 2y1 2x1 + 3y1 x = , y = − , (7.8) √13 √13 will rotate the x, y axes to positions given by the respective columns of P . (More generally, we can always arrange for the x1 axis to point either into the first or fourth quadrant.) 12 6 Now A = − is the matrix of the quadratic form 6 7 · − ¸ 12x2 12xy + 7y2, − so we have, by Theorem 7.1.5 2 2 2 2 12x 12xy + 7y = 16x1 + 3y1.
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