-5 -14 2 Consider the symmetric matrix A = -14 4 -16 2 -16 10 (a)...

 

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-5 -14 2 Consider the symmetric matrix A = -14 4 -16 2 -16 10 (a) Check that the following are eigenvectors for A, determining the eigenvalue for each (all eigenvalues are integers): V = , V = V3 = 2 (These vi's are readily seen to be nonzero and pairwise orthogonal, as the Spectral Theorem guaran- tees can always be arranged, so these v's constitute an orthogonal basis of R.) (b) Write each standard basis vector e; as a linear combination of V1, V2, V3 by computing the projection of e; onto each of V1, V2, V3. (c) Use the following strategy to compute A10 (you can leave expressions like 10, where A is an eigen- value of A, in your answer, rather than multiplying them all out): for each i = 1, 2, 3, the ith column of A10 is given by A0e, which we can compute by writing e, in the orthogonal basis of eigenvectors from part (a). As a safety check, make sure that your answer is symmetric, since any power of a symmetric matrix is symmetric (you might like to think on your own about why that is true). (d) Use another strategy to compute A10: since each v; has length 3, the eigenvectors v = vi/3 are unit vectors and hence constitute an orthonormal basis. Thus, the matrix Q with ith column v is orthogonal and we know that A = QDQ = QDQT where D is the diagonal 3 3 matrix with ith entry, so A10 QD10Q-1 = QD10QT. Write out Q and compute this expression for A10. (Hint: before multiplying matrices, you should be Activate Wir able to cancel out the 3's in the denominators, so all work is then with integers, not fractions.) You Go to Settings t should get the same answer as for (c)! -5 -14 2 Consider the symmetric matrix A = -14 4 -16 2 -16 10 (a) Check that the following are eigenvectors for A, determining the eigenvalue for each (all eigenvalues are integers): V = , V = V3 = 2 (These vi's are readily seen to be nonzero and pairwise orthogonal, as the Spectral Theorem guaran- tees can always be arranged, so these v's constitute an orthogonal basis of R.) (b) Write each standard basis vector e; as a linear combination of V1, V2, V3 by computing the projection of e; onto each of V1, V2, V3. (c) Use the following strategy to compute A10 (you can leave expressions like 10, where A is an eigen- value of A, in your answer, rather than multiplying them all out): for each i = 1, 2, 3, the ith column of A10 is given by A0e, which we can compute by writing e, in the orthogonal basis of eigenvectors from part (a). As a safety check, make sure that your answer is symmetric, since any power of a symmetric matrix is symmetric (you might like to think on your own about why that is true). (d) Use another strategy to compute A10: since each v; has length 3, the eigenvectors v = vi/3 are unit vectors and hence constitute an orthonormal basis. Thus, the matrix Q with ith column v is orthogonal and we know that A = QDQ = QDQT where D is the diagonal 3 3 matrix with ith entry, so A10 QD10Q-1 = QD10QT. Write out Q and compute this expression for A10. (Hint: before multiplying matrices, you should be Activate Wir able to cancel out the 3's in the denominators, so all work is then with integers, not fractions.) You Go to Settings t should get the same answer as for (c)!