Solved: Set C = triu(ones(4), 1) + diag([1, 1], 2) and [X, D] = eig(C) Compute X1CX and

Chapter 6, Problem 15

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Set C = triu(ones(4), 1) + diag([1, 1], 2) and [X, D] = eig(C) Compute X1CX and compare the result with D. Is C diagonalizable? Compute the rank of X and the condition number of X. If the condition number of X is large, the computed values for the eigenvalues may not be accurate. Compute the reduced row echelon form of C. Explain why 0 must be an eigenvalue of C and the corresponding eigenspace must have dimension 1. Use MATLAB to compute C4. It should equal the zero matrix. Given that C4 = O, what can you conclude about the actual values of the other three eigenvalues of C? Explain. Is C defective? Explain. If the computations had been done in exact arithmetic,the matrix B would be similar to A andhence defective. Use MATLAB to compute the eigenvaluesof B and a matrix X consisting of theeigenvectors of B. Determine the rank of X. Is thecomputed matrix B defective? Because of roundingerror, a more reasonable question to ask is whetherthe computed matrix B is close to being defective(i.e., are the column vectors of X close to beinglinearly dependent?). To answer this question, useMATLAB to compute rcond(X), the reciprocal ofthe condition number of X. A value of rcond closeto zero indicates that X is nearly rank deficient.