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hw3 - × 2 matrix that changes each time you select it Run...

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MATH 343:001 (11814) Applied Linear Algebra Fall 2009 Homework 3 Due Wednesday, October 21 Turn in answers to the following problems. Section 5.1: 1, 2, 5, 6, 21, 22, 36 Section 5.2: 2, 3, 4, 11, 13, 23, 24, 29, 30, 37 MATLAB PROBLEMS: (1) The MATLAB command eigshow offers a graphical demonstration of eigenvalues and singular values (as shown in class). For now focus on the eigenvalues mode, singular values will be discussed later in the course. On the top of the window there is a drop down menu with a long list of matrices to choose for the demonstration. For this problem look at entries 10, 11, and 12. Explore the animation to visualize the eigenvectors and eigenvalues. Use the MATLAB command eig to compute the eigenvalues and eigenvectors of each of the three matrices. Look carefully at the vectors returned by MATLAB. Are the matrices diagonalizable? (2) Refer again to the command eigshow . The last entry is a random 2
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Unformatted text preview: × 2 matrix that changes each time you select it. Run it 10 times and determine which percentage of the matrices had real eigenvalues. You can compute the eigenvalues with the eig command each time or determine if they are real by visual inspection (which is faster). (3) The MATLAB command magic(n) produces an n × n magic square: that is, a matrix of integers from 1 to n 2 such that each row, column and the two diagonals add up to the same number. Compute the eigenvalues of the magic matrices generated by MATLAB for n = 3, 4, and 5. What do you notice about the largest eigenvalue and its corresponding eigenvector (keep in mind that MATLAB produces eigenvectors of length 1, but any multiple of them will do)? What is your conjecture about the largest eigenvalue of magic(n) for all n ? Gerardo Lafferriere, October 15, 2009 1...
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