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m1_lec15_2up - 1 Review Last lecture we learnt how to...

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Unformatted text preview: 1 Review Last lecture we learnt how to orthogonally diagonalise a real symmetric matrix A. Quadratic Forms: Conic Sections 2 The equation: ax 2 + bxy + cy 2 + dx + ey + f = 0 can always be written in matrix form: x y a b/2 b/2 c x +d y e x +f =0 y or as xt Ax + K x + f = 0 where A= a b/2 b/2 c x= x y and K= d e. xt Ax is called a quadratic form in two variables. More generally, if A is a symmetric n × n (real) matrix and x ∈ Rn , xt Ax is a quadratic form in n variables. General procedure for identifying conics. 3 Case 1: K = [0 0]. (a) Write equation as xt Ax + f = 0 where A is real and symmetric. (b) Find P orthogonally diagonalising A and such that det(P ) = 1. (c) Substitute x = P x to transform equation to (x )t P t AP x + f = (x )t D x + f = λ1 (x )2 + λ2 (y )2 + f = 0 where D has diagonal entries λ1 and λ2 . (d) Rearrange into standard form to identify conic. Note: If P is orthogonal and det(P ) = 1 then P is a rotation: P= cos θ sin θ − sin θ . cos θ 4 Case 2: K ≠ [0 0]. Do (a), (b) and (c) as before to get: λ1 (x )2 + λ2 (y )2 + KP x + f = 0 or λ1 (x )2 + λ2 (y )2 + c x + d y + f = 0. (d) Complete the square to get λ1 (x − x0 )2 + λ2 (y − y0 )2 = f (e) Substitute x = x − x0 and y standard form to identify conic. = y = y0 and rearrange into ...
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