midterm-sampleFall04

# midterm-sampleFall04 - C-λI 4(5 points Let A be symmetric...

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18.335 Practice Midterm 1. (5 points) Let A be real symmetric and positive semideﬁnite, i.e. x T Ax 0 for all x 6 = 0. Show that if the diagonal of A is zero, then A is zero. 2. (5 points) Show that if Y = " I Z 0 I # then κ F ( Y ) = 2 n + k Z k 2 F . 3. Let T = a 1 b 1 c 1 . . . . . . . . . . . . b n - 1 c n - 1 a n be a real, n -by- n , nonsymmetric tridiagonal matrix where c i b i > 0 for all 1 i n - 1. Show that the eigenvalues of T are real (5 points) and distinct (5 points). Hint: Find a diagonal matrix D such that C = DTD - 1 is symmetric. Then argue about the rank of
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Unformatted text preview: C-λI . 4. (5 points) Let A be symmetric positive deﬁnite matrix with Cholesky factor C , i.e. A = C T C . Show that k A k 2 = k C k 2 2 . 5. (5 points) If A and B are real symmetric positive deﬁnite matrices then decide whether the following are true, justifying your results: • A + B is symmetric positive deﬁnite. • A · B is symmetric positive deﬁnite. 6. (5 points) Prove that det( I + xy T ) = 1 + x T y . 1...
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