Assignment 8 solution

Assignment 8 solution - MATH 223, Linear Algebra Winter,...

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Unformatted text preview: MATH 223, Linear Algebra Winter, 2008 Solutions to Assignment 8 1. Calculate the following; the inner product space is C 3 , with the standard inner product. < 1 1- i 3- 2 i , - 2 i 3- 7 i 4- i > , < 1- i 2 + i 3 , 1- i 2 + i 3 > , < 2 + i 1- i , 2- i 1 + i > . Solution: The first one is 1(+2 i ) + (1- i )(3 + 7 i ) + (3- 2 i )(4 + i ) = 2 i +10+4 i +14- 5 i = 24+ i . The second is (1- i )(1+ i )+(2+ i )(2- i )+3(3) = 2 + 5 + 9 = 16. (Unsurprisingly, this one comes out as a positive real number.) The last is (2+ i )(2+ i )+(1- i )(1- i )+0(0) = 3+4 i- 2 i +0 = 3 + 2 i . 2. Suppose that A is a Hermitian matrix and that it has an eigenvalue 0. Show that A is not positive definite; that is, there is a nonzero vector ~v such that ~v T A ~v 0. Solution: Let ~w be an eigenvector corresponding to and ~v = ~w . So A ~v = A~w = ~w . ~v T A ~v = ~w T ( ~w ) = || ~w || 2 . Since || ~w || > 0, this has the same sign as . 3. For each of the following matrices A j , decide whether the function on C 3 C 3 defined by < ~v, ~w > = ~v T A j ~w is an inner product on C 3 ....
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Assignment 8 solution - MATH 223, Linear Algebra Winter,...

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