Problem_Set2 - EEL 3105 Fall 2011 Problem Set 2 1. Consider...

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Unformatted text preview: EEL 3105 Fall 2011 Problem Set 2 1. Consider the vector v [1 2.8 3 3.33] Find vvT. Is it symmetric? Find its eigenvalues. Is it invertible? 2. 3. 4. 5. Let v be an nx1 vector. Find eigenvalues of vvT. For choices of v is vvT invertible? Let A and B be two square nxn matrices. Find a formula for (A+B)2. Suppose AB=BA. Find a formula for (A+B)n. What will happen if A and B do not commute? Suppose the characteristic polynomial of a 3x3 matrix A is s 3 4 s 2 5s 3 Find the characteristic polynomial of I+A where I is a 3x3 identity matrix. 6. Suppose A is an nxn matrix with characteristic polynomial s n n 1s n 1 n 2 s n 2 1s 0 Find the characteristic polynomial of I+A where I is an nxn identity matrix. 7. Let A and B be two matrices such that A is nxm and B is mxn. Is AB+BTAT is a symmetric matrix? Please justify your answer. 8. Let A be a symmetric matrix. Suppose 1+jb is an eigenvalue of A. Find b. 9. Consider the matrix 1 b A b 2 Let v = [1 ] be a 1x2 row vector. A. Calculate vAvT. B. Set =‐1. Under what conditions on b is vAvT positive? C. Find eigenvalues of A. Under what conditions on b are both eigenvalues positive? 10. Let 1 2 A 4 5 3 1 Find eigenvalues of B= ATA. Are they positive? Find the corresponding eigenvectors w1 and w2. Calculate T T T T w1 Bw1 and w2 Bw2 . Calculate their ratio to w1 w1 and w2 w2 respectively. Comment on your answer. 11. Suppose A is an nxm matrix. Set B = ATA. Suppose is an eigenvalue of B. Suppose w is an eigenvector of B corresponding to the eigenvalue. Find a formula for wTBw. Can you use this to show that cannot be negative? ...
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This note was uploaded on 12/27/2011 for the course EEL 3105 taught by Professor Boykins during the Fall '10 term at University of Florida.

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