HOMEWORK 1 - DREXEL UNIVERSITY Department of Mechanical...

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DREXEL UNIVERSITY Department of Mechanical Engineering & Mechanics Applied Engineering Analytical & Numerical Methods I MEM 591 - Fall 2013 HOMEWORK #1: Due Thursday, October 3 rd (6:15PM in class) 1. [30 points] i) Let Q 1 , Q 2 2 א ܴ ௡௫௡ be unitary matrices. Prove that Q 1 Q 2 is also unitary. ii) Given any A א ܴ ௡௫௡ , its eigenpair ( λ ,v) and what is known as its similarity transformation B = XAX -1 (for any X א ܴ ௡௫௡ ), show that B has the same eigenvalues as A . Compute also the eigenvectors of B . iii) Let ݔ א ܴ . Prove that ԡݔԡ ൑ √݊ ԡݔԡ 2. [20 points] Consider the matrix ܣ ൌ ൭ െ20 െ5 6 െ5 5 12 6 12 െ8 Compute by hand its eigenvalues and eigenvectors for the case of orthonormal eigenvectors.
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Unformatted text preview: 3. [20 points] (i) Prove that ܣ ் ܣ is symmetric and also that is positive semi-definite. (ii) Consider the matrix ܣ ൌ ൭ െ2 1 2 1 െ1 ൱ Compute by hand, a singular value decomposition of A. Compare with the results you obtain in MATLAB. 4. [30 points] Construct in MATLAB the following subroutines. Compare your code results with the ones you obtain by using built-in commands in MATLAB and report your findings. i) Inner product of any two vectors. ii) Any Matrix-Matrix multiplication. iii) Any Vector 2-norm computation....
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  • Fall '10
  • KONTSOS
  • Linear Algebra, Matrices, Orthogonal matrix, Applied Engineering Analytical & Numerical Methods, similarity transformation B=XAX-1

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