Quiz 8 - Solutions

# Quiz 8 - Solutions - Math 54 Quiz 8 Mike Hartglass November...

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Math 54 Quiz 8 Mike Hartglass November 6, 2010 1.) Consider the matrix A = ± 1 3 3 1 ² . Find matrices Q and D where Q is an or- thogonal matrix, D is a diagonal matrix and A = QDQ - 1 . The characteristic polynomial is ( t - 1) 2 - 9 so its roots are - 2 and 4. For t = 2, A - tI = ± 3 3 3 3 ² which is easily seen to have null space generated by (1 , - 1) T . Similarly, the null space for A - 4 I is generated by the vector (1 , 1) T . Normalizing these vectors (as we require Q to be an orthogonal matrix) we see that Q = ± 1 2 1 / 2 1 / 2 - 1 2 ² and D = ± 4 0 0 2 ² . 1

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2.) Let V be the space C [ - 1 , 1] with inner product h f,g i = ´ 1 - 1 f ( x ) g ( x ) dx . Find an orthogonal basis for the subspace spanned by the polynomials 1,
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Quiz 8 - Solutions - Math 54 Quiz 8 Mike Hartglass November...

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