Math 307 Spring, 2005 Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-mail: hentzel@iastate.edu http://www.math.iastate.edu/hentzel/class.307.05 Text: Linear Algebra and its Applications, Lay ============================================================ Final Exam 307 Matrix Theory Wednesday, May 4, 2005 9:45-11:45 AM Friday, April 29 7.2 Quadratic forms Main Idea: Remove the xy terms from degree two polynomials. T Key Words: Quadratic Forms, X A X, X = P Y 2 2 2 Goal: Given Ax + By + Cz + Dxy + Exz + Fyz + G = 0 find an equivalent equation of the form 2 2 2 A'u + B'v + C'w + D' = 0 Example: Make a change a basis with orthogonal matrix P | x | | u | | y | = P | v | | z | | w | that eliminates the cross product terms. 2 2 2 17 x + 11 y + 17 z + 8 xy - 16 xz -8 yz = 9 Theory: We Factor the equation as T X A X = 9 | 17 4 -8 | | x | (x,y,z) | 4 11 -4 | | y | = 9 | -8 -4 17 | | z | If you multiply out this left hand side, you get the original equation. You split the cross product terms evenly because it is important to have a symmetric matrix to work with. Making the substituion T X = PY into X A X = 9 gives T (PY) A (PY) = 9 or T T Y P A P Y = 9. Choosing P orthogonal so that T | A' 0 0 | P A P = | 0 B' 0 | | 0 0 C'| The new equation is 2 2 2 A' u + B' v + C' w = 9 So essentially, we now do the process to find an orthogonal matrix P such that -1 P A P is diagonal. And since P is orthogonal, the fact that we T -1 wanted P A P and computed P A P will cause no problem because the transpose is the inverse! |17 4 -8| A = | 4 11 -4| |-8 -4 17| Eigenvalues[A] = {9, 9, 27} Eigenvectors[A] x=0 x=0 x=27 1 -1 -2 0 2 -1 1 0 2 The first two vectors are not orthogonal. Must use Gram-Schmidt. | 1 | |-1 | | 0 | | 2 | | 1 | | 0 | V1 V2 | 1 | W1 = | 0 | | 1 | V2oW1 | -1 | -1 | 1 | |-2+1| |-1| |-1| W2 = V2 - ------- W1 = | 2 | - ----| 0 | = 1/2 | 4+0| = 1/2| 4| USE| 4| W1oW1 | 0 | 2 | 1 | | 0+1| | 1| | 1| 1 -1 -2 | 1/Sqrt[2] -1/Sqrt[18] -2/3 | P = 0 4 -1 = | 0 4/Sqrt[18] -1/3 | 1 1 2 | 1/Sqrt[2] 1/Sqrt[18] 2/3 | ------- ------ -------- Sqrt[2] Sqrt[18] 3 T | 9 0 0 | P A P = | 0 9 0 | | 0 0 27 | The new equation is 2 2 2 9u + 9 v + 27 w = 9 2 2 2 u v w ---- + ----- + ---- = 1 1 1 1/3 The surface is an ellipsoid. The absolute value of u and v cannot be larger than 1 and the absolute value of w can not be larger than 1/Sqrt[3]. The new coordinate system will be the columns of P That is, the u v w axes will be along | 1 | |-1 | |-2 | | 0 | | 4 | |-1 | | 1 |, | 1 |,| 2 | We will examine the surface from these axes. ---------------------------------------------------------- 2 2 2 17 x + 11 y + 17 z + 8 xy - 16 xz -8 yz = 9 X = PU so U = Transpose[P].X ------------------------------------------------------------ A = {{17, 4,-8}, { 4,11,-4}, {-8,-4,17}}; P = {{ 1/Sqrt[2], -1/Sqrt[18], -2/3 }, { 0 , 4/Sqrt[18], -1/3 }, { 1/Sqrt[2], 1/Sqrt[18], 2/3 }}; Q = Transpose[P]; f[x_,y_,z_] := 17 x^2 + 11 y^2 + 17 z^2 + 8 x y -16 x z - 8 y z; ans = Solve[ f[x,y,z] == 9,z]; s1[x_,y_] = z /. ans[[1]]; s2[x_,y_] = z /. ans[[2]]; p1 = Plot3D[ s1[x,y],{x,- 1, 1},{y,-1, 1},PlotPoints->50]; p2 = Plot3D[ s2[x,y],{x,- 1, 1},{y,-1, 1},PlotPoints->50]; p3 = Graphics3D[ {Thickness[0.02],RGBColor[1,0,0],Line[{ -2 Q[[1]],2 Q[[1]] }] } ]; p4 = Graphics3D[ {Thickness[0.02],RGBColor[0,1,0],Line[{ -2 Q[[2]],2 Q[[2]] }] } ]; p5 = Graphics3D[ {Thickness[0.02],RGBColor[0,0,1],Line[{ -2 Q[[3]],2 Q[[3]] }] } ]; p6 = Graphics3D[ { PointSize[0.02], Point[ Q[[1]] ],Point[ Q[[2]] ],Point[ Q[[3]]/Sqrt[3] ], Point[-Q[[1]] ],Point[-Q[[2]] ],Point[-Q[[3]]/Sqrt[3] ]}]; p7 = Show[p1,p2,p3,p4,p5,p6,AspectRatio-> Automatic, PlotLabel->"17x^2+11y^2+17z^2+8xy-16xz-8yz=9"]; rag = ParametricPlot3D[ P.{r Cos[t],r Sin[t], Sqrt[1/3 (1-r^2)]},{t,0,2 Pi},{r,0,1}, PlotPoints->50]; tag = ParametricPlot3D[ P.{r Cos[t],r Sin[t],-Sqrt[1/3 (1-r^2)]},{t,0,2 Pi},{r,0,1}, PlotPoints->50]; p8 = Show[rag,tag,p3,p4,p5,p6,PlotLabel->"17 x^2 + 11 y^2 + 17 z^2 + 8 x y -16 x z - 8 y z=9", AspectRatio->Automatic]; Display["quadratic.ps",p8]; ---------------------------------------------------------------------------