Lecture_11_2 - Lecture Note 11-2: Nov 21 - Nov 24, 2006 Dr....

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Unformatted text preview: Lecture Note 11-2: Nov 21 - Nov 24, 2006 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong jwong@math.cuhk.edu.hk MAT 2310 Linear Algebra and Its Applications Fall, 2006 Produced by Jeff Chak-Fu WONG 1 Q UADRATIC F ORMS QUADRATIC FORMS 2 In your precalculus and calculus courses you have seen that the graph of the equation ax 2 + 2 bxy + cy 2 = d, (1) where a,b,c and d are real numbers, is a conic section centered at the origin of a rectangular Cartesian coordinate system in two-dimensional space. Similarly, the graph of the equation ax 2 + 2 dxy + 2 exy + by 2 + 2 fyz + cz 2 = g, (2) where a,b,c,d,e,f and g are real numbers, is a quadratic surface centered at the origin of a rectangular Cartesian coordinate system in three-dimensional space. If a conic section or quadratic surface is not centered at the origin, its equations are more complicated than those given in (1) and (2). QUADRATIC FORMS 3 The identification of the conic section or quadratic surface that is the graph of a given equation often requires the rotation and translation of the coordinate axes . These methods can best be understood as an application of eigenvalues and eigenvectors of matrices. The expressions on the left sides of Equations (1) and (2) are examples of quadratic forms. Quadratic forms arise • in statistics, mechanics and in other problems in physics; • in quadratic programming; • in the study of maxima and minima of functions of several variables; and in other applied problems. In this section we use our results on eigenvalues and eigenvectors of matrices to give a brief treatment of real quadratic forms in n variables. QUADRATIC FORMS 4 DEFINITION If A is a symmetric matrix, then the function g : R n → R 1 (a real-valued function on R n ) defined by g ( x ) = x T A x , where x = x 1 x 2 . . . x n , is called a real quadratic form in the variables x 1 , x 2 , ... , x n . The matrix A is called the matrix of the quadratic form g . We shall also denote the quadratic form by g ( x ) . QUADRATIC FORMS 5 Example 1 Write the left side of (1) as the quadratic form in the variables x and y . Solution Let x = x y and A = a b b c . Then the left side of (1) is the quadratic form g ( x ) = x T A x . QUADRATIC FORMS 6 Example 2 Write the left side of (2) as the quadratic form. Solution Let x = x y z and A = a d e d b f e f c . Then the left side of (2) is the quadratic form g ( x ) = x T A x . QUADRATIC FORMS 7 Example 3 The following expressions are quadratic forms: (a) 3 x 2- 5 xy- 7 y 2 = [ x y ] 3- 5 2- 5 2- 7 x y (b) 3 x 2- 7 xy +5 xz +4 y 2- 4 yz- 3 z 2 = [ x y z ] 3- 7 2 5 2- 7 2 4- 2 5 2- 2- 3 x y z QUADRATIC FORMS 8 The other form writes as follows: g ( x ) = n X i =1 n X j =1 b ij...
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Lecture_11_2 - Lecture Note 11-2: Nov 21 - Nov 24, 2006 Dr....

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