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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 λ ≤ 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 λ and ~v = ~w . So A ~v = A~w = λ~w . ~v T A ~v = ~w T ( λ~w ) = λ  ~w  2 . Since  ~w  > 0, this has the same sign as λ . 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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This note was uploaded on 04/29/2008 for the course MATH 223 taught by Professor Loveys during the Fall '07 term at McGill.
 Fall '07
 Loveys
 Linear Algebra, Algebra

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