q1sols18085f04

q1sols18085f04 - MIT OpenCourseWare http/ocw.mit.edu 18.085...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 18.085 Quiz 1 October 4, 2004 Professor Strang Your name is: SOLUTIONS Grading 1. 2. 3. 4. OPEN BOOK EXAM Write solutions onto these pages ! Circles around short answers please !! 1) (32 pts.) This problem is about the symmetric matrix ⎤ ⎡ 2 − 1 H = ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ − 1 2 − 1 − 1 1 (a) By elimination find the triangular L and diagonal D in H = LDL T . H (b) What is the smallest number q that could replace the corner entry 33 = 1 and still leave H positive semi definite ? q = (c) H comes from the 3-step framework for a hanging line of springs: A C A T displacements −→ elongations −→ spring forces −→ external force f What are the specific matrices A and C in H = A T CA ? (d) What are the requirements on any m by n matrix A and any symmetric matrix C for A T CA to be positive definite ? ⎤ ⎡ ⎤ ⎡ 2 − 1 2 − 1 ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ −→ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ 1....
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q1sols18085f04 - MIT OpenCourseWare http/ocw.mit.edu 18.085...

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