looking for part D/E , already asked for A, B C, in prev. question
In this exercise, we will explore the properties of anti-symmetric matrices. A matrix M is called "anti- symmetric" if it satisfies MT-M (a) Create a 6 by 6 anti-symmetric matrix by the following steps: (1) input the 6x6 matrix A into MATLAB 2 0 1 7 0 3 5 4 1 8 9 8 3 6 7 1 5 9 0 1 0 2 9 4 5 3 2 4 6 8 0 7 8 6 3 4 Set B A-AT. Now you have an anti-symmetric matrix B. Report B (b) Report the eigenvalues of your anti-symmetric matrix (c) What do you notice about these eigenvalues? (Please remember that the order in which the computer gives you the eigenvalues is arbitrary- it could just as easily give the eigenvalues to you in a different order. You are looking for patterns in the numbers themselves, not in the particular ordering that the computer has given them in) (d) Are the eigenvectors of your anti-symmetric matrix orthogonal at least up to machine precision? OK it is surely tedious to check pair by pair in MATLAB. Yes who wants to do that...Try to think of easier ways to do the calculation, by using a for loop or manipulating the matrix consisting of all the eigenvalues You don't need to hand in your codes, but just answer yes or no. (Please note that this is a question about the eigenvectors- not the antisymetric matrix itself.) (e) Create a number of 5x5 anti-symmetric matrices arbitrarily by yourself (Don't submit your matrices) You might want to use the rand function in MATLAB and then follow the steps outlined in 3(a). What do the eigenvalues of these matrices look like (Don't submit the eigenvalues)? What about 7x7, 8x8 or 32 x32 (Don't submit this)? What changes in the eigenvalues as you change the size of your matrix? What stays the same? Write down one to two sentences describing any pattern that you see as you vary the size of the matrix. In this exercise, we will explore the properties of anti-symmetric matrices. A matrix M is called "anti- symmetric" if it satisfies MT-M (a) Create a 6 by 6 anti-symmetric matrix by the following steps: (1) input the 6x6 matrix A into MATLAB 2 0 1 7 0 3 5 4 1 8 9 8 3 6 7 1 5 9 0 1 0 2 9 4 5 3 2 4 6 8 0 7 8 6 3 4 Set B A-AT. Now you have an anti-symmetric matrix B. Report B (b) Report the eigenvalues of your anti-symmetric matrix (c) What do you notice about these eigenvalues? (Please remember that the order in which the computer gives you the eigenvalues is arbitrary- it could just as easily give the eigenvalues to you in a different order. You are looking for patterns in the numbers themselves, not in the particular ordering that the computer has given them in) (d) Are the eigenvectors of your anti-symmetric matrix orthogonal at least up to machine precision? OK it is surely tedious to check pair by pair in MATLAB. Yes who wants to do that...Try to think of easier ways to do the calculation, by using a for loop or manipulating the matrix consisting of all the eigenvalues You don't need to hand in your codes, but just answer yes or no. (Please note that this is a question about the eigenvectors- not the antisymetric matrix itself.) (e) Create a number of 5x5 anti-symmetric matrices arbitrarily by yourself (Don't submit your matrices) You might want to use the rand function in MATLAB and then follow the steps outlined in 3(a). What do the eigenvalues of these matrices look like (Don't submit the eigenvalues)? What about 7x7, 8x8 or 32 x32 (Don't submit this)? What changes in the eigenvalues as you change the size of your matrix? What stays the same? Write down one to two sentences describing any pattern that you see as you vary the size of the matrix
