A8_soln - Math 235 Assignment 8 Due Wednesday July 14th 1...

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Unformatted text preview: Math 235 Assignment 8 Due: Wednesday, July 14th 1. Sketch the graph of 9 x 2 + 4 xy + 6 y 2 = 21 showing both the original and new axes. Solution: The corresponding symmetric matrix is bracketleftbigg 9 2 2 6 bracketrightbigg . We find that the characteristic polynomial is C ( λ ) = vextendsingle vextendsingle vextendsingle vextendsingle 9- λ 2 2 6- λ vextendsingle vextendsingle vextendsingle vextendsingle = λ 2- 15 λ + 50 = ( λ- 10)( λ- 5) . So, we have eigenvalues λ 1 = 10 and λ 2 = 5. For λ 1 = 10 we get A- λ 1 I = bracketleftbigg- 1 2 2- 4 bracketrightbigg ∼ bracketleftbigg- 1 2 bracketrightbigg . Thus, a corresponding eigenvector is vectorv 1 = bracketleftbigg 2 1 bracketrightbigg . For λ 2 = 5 we get A- λ 2 I = bracketleftbigg 4 2 2 1 bracketrightbigg ∼ bracketleftbigg 2 1 0 0 bracketrightbigg . Thus, a corresponding eigenvector is vectorv 2 = bracketleftbigg 1- 2 bracketrightbigg . Thus, we have the ellipse 10 x 2 1 + 5 x 2 2 = 21 with principal axis vectorv 1 for x 1 and vectorv 2 for y 1 . Graphing gives: 2 2. Find a singular value decomposition of 1- 4- 2 2 2 4 .....
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

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A8_soln - Math 235 Assignment 8 Due Wednesday July 14th 1...

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