WINTER 2007 assign9-sol

WINTER 2007 assign9-sol - Math 235 Assignment 9 Due...

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Unformatted text preview: Math 235 Assignment 9 Due 9:15am, Wednesday March 28, 2007. From the Text § 7.1 #34. Construct a spectral decomposition of the matrix A that has the orthogonal diagonalization: 3- 2 4- 2 6 2 4 2 3 = 1 √ 2- 1 √ 18- 2 3 4 √ 18- 1 3 1 √ 2 1 √ 18 2 3 7 7- 2 1 √ 2 1 √ 2- 1 √ 18 4 √ 18 1 √ 18- 2 3- 1 3 2 3 . Solution. Denote the columns of P by u 1 , u 2 and u 3 . Then A = 7 u 1 u T 1 + 7 u 2 u T 2- 2 u 3 u T 3 To verify this decomposition of A, compute u 1 u T 1 = 1 √ 2 1 √ 2 1 √ 2 1 √ 2 = 1 2 1 2 1 2 1 2 u 2 u T 2 = - 1 √ 18 4 √ 18 1 √ 18 - 1 √ 18 4 √ 18 1 √ 18 = 1 18- 4 18- 1 18- 4 18 16 18 4 18- 1 18 4 √ 18 1 18 u 3 u T 3 = - 2 3- 1 3 2 3 - 2 3- 1 3 2 3 = 4 9 2 9- 4 9 2 9 1 9- 2 9- 4 9- 2 9 4 9 and 7 u 1 u T 1 + 7 u 2 u T 2- 2 u 3 u T 3 = 7 2 7 2 7 2 7 2 + 7 18- 28 18- 7 18- 28 18 112 18 28 18- 7 18 28 √ 18 7 18 - 8 9 4 9- 8 9 4 9 2 9- 4 9- 8 9- 4 9 8 9 = 3- 2 4- 2 6 2 4 2 3 ....
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This note was uploaded on 03/03/2010 for the course MATH 235 taught by Professor Celmin during the Winter '08 term at Waterloo.

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WINTER 2007 assign9-sol - Math 235 Assignment 9 Due...

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