6 Eigenvalue/Eigenvector Problems (I) 1. Write Maple and/or Matlab code that implements the Faddeev-Leverrier method seen in class. Use your code to compute the characteristic polynomial of the Hilbert matrix Hs by the Faddeev-Leverrier method. Compare your results with the char- acteristic polynomial as computed by built-in Maple or Matlab commands. 2. Prove that Hs has five positive eigenvalues. Denote these eigenvalues by , 2, 3, A, As Find numerical values for ,5 by using a numerical method of your choice (from the ones seen in class) to solve the characteristic polynomial. Use the numerical values you found for A to verify numerically that XA -563/315 aid that .-1/266716800000 3. Prove that Hs is a positive-definite matrix. 4. Since Hs is symmetric and positive-definite, it possesses a Cholesky decomposition Hs = L.LT Compute the lower triangular matrix L by writing 2 0 0 0 s2 00 41 42 143 144 0 L 54 55 and subsequently performing the multiplication L LT and equating the elements of the resulting matrix with the corresponding elements of Hs. Compare your result with the result found by built-in Maple or Matlab commands. 5. Solve the following five systems of linear equations: Hk-z-i, k-6, 7, 8, 9. 10. where is the k 1 column vector whose all k entries are equal to 1, for k = 6,7,8,9, 10. Examine the solutions closely and mention any kind of patterns that you may notice If you noticed any pattern in the question above, can will in fact occur for every k? you prove that this pattern