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Unformatted text preview: A is positive deﬁnite, then A − 1 is symmetric and positive deﬁnite. (c) Classify the following matrices according to whether they are positive or negative deﬁnite or semideﬁnite or indeﬁnite: ( 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 semideﬁnite. 5. Deﬁne 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
 Linear Algebra, Matrices, Eigenvalues, Orthogonal matrix, λI

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