Solved: Set C = ones(6) + 7 eye(6) and [X, D] = eig(C) (a) Even though = 7 is an

Chapter 6, Problem 23

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Set C = ones(6) + 7 eye(6) and [X, D] = eig(C) (a) Even though = 7 is an eigenvalue of multiplicity 5, the matrix C cannot be defective. Why? Explain. Check that C is not defective by computing the rank of X. Compute also XT X. What type of matrix is X? Explain. Compute also the rank of C 7I. What can you conclude about the dimension of the eigenspace corresponding to = 7? Explain. (b) The matrix C should be symmetric positive definite. Why? Explain. Thus, C should have a Cholesky factorization LLT . The MATLAB command R = chol(C) will generate an upper triangular matrix R that is equal to LT . Compute R in this manner and set L = R . Use MATLAB to verify that C = LLT = RTR (c) Alternatively, one can determine the Cholesky factors from the LU factorization of C. Set [ L U ] = lu(C) and D = diag(sqrt(diag(U))) and W = (L D) How do R and W compare? This method of computing the Cholesky factorization is less efficient than the method MATLAB uses for its Chol function.