Week13_VP1 - Week XIII Application I Quadratic Forms A...

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Week XIII : Application I. Quadratic Forms A general quadratic form in two variables is 22 (, ) . f xy a x b x y c y ±± This can be put in the form >@ 2 or 2 T ab x x yx bc y ªº ª º «» « » ¬¼ ¬ ¼ Ax G G where A is the 2 u 2 symmetric matrix above. For three variables we have 222 ad e x ax by cz dxy exz fyz x y z d b f y ef c z . ª ºª º « »« » ±±±±± « « ¬ ¼¬ ¼ Now, the symmetric matrix of the quadratic form is of order 3. The general form in n variables can be written as T x Ax G G , where > @ 12 ,, . . . , T n x xx x G and A is a symmetric n u n matrix with the coefficients of the squared term down the diagonal and half the coefficients of x i x j in the ij th and the ji th entries of A . Since A is symmetric, there is an orthogonal matrix Q that will diagonalize it. Thus, Q -1 AQ = Q T AQ = D , where the columns of Q are the orthonormal set of eigenvectors of A and the diagonal matrix D contains the eigenvalues of A . The quadratic equation T x Ax G G = 1 can then be written as TT x QDQ x G G = 1 or ² ³ ² ³ T Qx DQx G G = 1. If we define x Qt G G as a transformation from the coordinate system to the coordinate system, then the inverse transformation is given by (, , . . . ) tt . . . ) . T tQ x G G In the coordinate system, the quadratic equation becomes . . . ) 1. T t G G Since D is diagonal, let us denote D = diag[ O 1 , O 2 ,…], then the quadratic equation has the canonical form
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22 11 2 2 1.
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Week13_VP1 - Week XIII Application I Quadratic Forms A...

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