Linear Algebra Examples C-4

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Com 37 Linear Algebra Examples c-4 5. 5 Find the type and position of the conical surface, which is given by the equation x2 + 4y 2 + z 2 + 20yz + 26zx + 20xy − 24 = 0. The corresponding matrix is ⎛ ⎞ 1 10 13 A = ⎝ 10 4 10 ⎠ 13 10 1 of the characteristic polynomial det(A − λI) = 1−λ 10 13 1 10 13 = −(λ − 24) 10 4−λ 10 1 4−λ 10 13 10 1−λ 1 10 1−λ = 24 − λ 24 − λ 10 4−λ 13 10 = −(λ − 24) 24 − λ 10 1−λ 1 0 10 −6 − λ 13 −3 0 0 −12 − λ = −(λ − 24)(λ + 6)(λ + 12). The three eigenvalues are λ1 = −12, λ2 = −6 and λ3 = 24.

Conical surfaces ⎛ ⎞ ⎛ −4 2 4 2 A − λ2 I = ⎝ 2 −1 −2 ⎠ ∼ ⎝ 0 4 −2 −4 0 ⎞ −1 −2 0 0 ⎠. 0 0 Two linearly independent eigenvectors are and v3 = (0, 2, −1), v2 = (1, 0, 1) √ where v2 = 2 and (by Gram-Schmidt) 1 1 (v3 · v2 ) v2 = (0, 2, −1) − (−1) · (1, 0, 1) = (1, 4, −1). 2 2 √ √ Since (1, 4, −1) = 18 = 3 2, the orthonormed eigenvectors are v3 = 1 v2 2 1 q2 = √ (1, 0, 1) 2 and 1 q3 = √ (1, 4, −1). 3 2 Sharp Minds - Bright Ideas! Please click the advert Employees at FOSS Analytical A/S are living proof of the company value - First - using new inventions to make dedicated solutions for our customers.

The axis of rotation is the Y1 -axis. com 58 Linear Algebra Examples c-4 6. 6 Given a conical surface in ordinary rectangular coordinates in space by the equation 4xy + az 2 = 1, where a ∈ R. 1. Find a quadratic form λ1 x21 + λ2 y12 + λ3 z12 , which the quadratic form, occurring on the left hand side of the equation can be reduced to by an application of some orthogonal substitution. (The orthogonal substitution is not requested). 2. Find all a, for which the given conical surface is a rotational surface and indicate for these values the type of the surface and a parametric description of its axis of rotation in the given coordinates x, y, z.

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