# By linearity b cos sin sin cos must rotate every

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Unformatted text preview: re 54 A= . 45 2.1.2. Find a 2 × 2-matrix B such that B is orthogonal and B −1 AB is diagonal, where 34 A= . 49 35 2.1.3. Find a 3 × 3-matrix B such that B is orthogonal and B −1 AB is diagonal, where 540 A = 4 5 0 . 001 2.1.4. Find a 3 × 3-matrix B such that B where −1 A= 2 0 is orthogonal and B −1 AB is diagonal, 20 0 2 . 21 2.1.5. Show that the n × n identity matrix I is orthogonal. Show that if B1 and B2 are n × n orthogonal matrices, so is the product matrix B1 B2 , as well as the − inverse B1 1 . 2.2 Conic sections and quadric surfaces The theory presented in the previous section can be used to “rotate coordinates” so that conic sections or quadric surfaces can be put into “canonical form.” A conic section is a curve in R 2 which is described by a quadratic equation, such as the equation ax2 + 2bx1 x2 + cx2 = 1, 1 2 (2.3) where a, b and c are constants, which can be written in matrix form as x1 If we let A= x2 ab bc ab bc and x1 x2 = 1. x= x1 x2 , we can rewrite (2.3) as xT Ax = 1, (2.4) where A is a symmetric matrix. According to the Spe...
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## This document was uploaded on 01/12/2014.

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