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Unformatted text preview: A is positive denite, then A 1 is symmetric and positive denite. (c) Classify the following matrices according to whether they are positive or negative denite or semidenite or indenite: ( a ) 1 0 0 0 3 0 0 0 5 . ( b ) 3 1 2 1 5 3 2 3 7 . ( c ) 2 4 4 8 3 . 1 4. Let A be a square n n matrix. (a) Show that A + A T is symmetric. (b) Show that x T Ax = x T ( A + A T 2 ) x for all x R n . Conclude that x T Ax 0 for all x R n if and only if the symmetric matrix A + A T is positive semidenite. 5. Dene f : R 2 R by setting f (0) = 0 and f ( x, y ) = xy x 2 + y 2 if ( x, y ) = 0 . For which vectors d = 0 does f (0; d ) exist? Evaluate it when it exists. 2...
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 Spring '09

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