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1. Let A = Onxn, where l> 1 is some integer. Charac- terize the eigenvalues of A. Explain your answer.

1. Let A = Onxn, where l> 1 is some integer. Charac- terize the eigenvalues of A. Explain your answer.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Let A = Onxn, where I ≥ 1 is some integer. Charac- terize the eigenvalues of A. Explain your answer. 2. Let A2 = A, where >1 is some integer. Character- ize the eigenvalues of A. Explain your answer. 3. Let A3x3 be real and satisfy A = -A. Explain, without calculating any determinants, why A is sin- gular (Hint: Explain why one of the eigenvalues must be zero. If so, what does this say about Ker(A)?). HE 4. Let Xmxn and Ynxm be given, with n ≥ m. State pre- cisely how the eigenvalues of XY and YX are related. Use this to find the eigenvalues of B = (i-j)i,j=1,...,n - (12) Find the spectral decomposition of A. In other words, diagonalize A via a real orthogonal matrix. 5. Let A = 1 1 6. Let A = -12 +420 01 Find the singular value decompo- sition of A. What is the best rank one approximation to A? 7. Let z, w E C. Show by direct calculation that the ( is normal. matrix M = -W Z 8. Use the Cayley Hamilton theorem to compute the in- 21 Explain your work. 1 3 verse of hp
Transcribed Image Text:1. Let A = Onxn, where I ≥ 1 is some integer. Charac- terize the eigenvalues of A. Explain your answer. 2. Let A2 = A, where >1 is some integer. Character- ize the eigenvalues of A. Explain your answer. 3. Let A3x3 be real and satisfy A = -A. Explain, without calculating any determinants, why A is sin- gular (Hint: Explain why one of the eigenvalues must be zero. If so, what does this say about Ker(A)?). HE 4. Let Xmxn and Ynxm be given, with n ≥ m. State pre- cisely how the eigenvalues of XY and YX are related. Use this to find the eigenvalues of B = (i-j)i,j=1,...,n - (12) Find the spectral decomposition of A. In other words, diagonalize A via a real orthogonal matrix. 5. Let A = 1 1 6. Let A = -12 +420 01 Find the singular value decompo- sition of A. What is the best rank one approximation to A? 7. Let z, w E C. Show by direct calculation that the ( is normal. matrix M = -W Z 8. Use the Cayley Hamilton theorem to compute the in- 21 Explain your work. 1 3 verse of hp
9. Consider the system of equations x=y=0; 2x - y = -1; 5x + y = 0 This system is evidently unsolvable. Write the normal equations for finding a least squares solution to it. Solve the normal equations. 0 2 0 10. Let A = 2 4 1 - 0-1 0 Compute its Schur-triangularization. Explain your work. (Hint; there is an obvious eigenvalue. Find an eigenvector; use cross- products to complete to an ONB for R³, etc., ) 11. Let Amxn and Bmxn be real matrices related via B = SAT, where S is mxm real orthogonal and T is nxn real orthogonal. Show that A and B have the same singular values and hence the same spectral norms. 12. Consider the system of equations 2x1 + x2 = 0, 0x1 + 0x2= 1, 1 + 2x2 = -1, Let A be the coefficient matrix of this system of equations. Find its SVD. Use that to find the Moore-Penrose inverse of A and thereby find the least squares solution to the system.
Transcribed Image Text:9. Consider the system of equations x=y=0; 2x - y = -1; 5x + y = 0 This system is evidently unsolvable. Write the normal equations for finding a least squares solution to it. Solve the normal equations. 0 2 0 10. Let A = 2 4 1 - 0-1 0 Compute its Schur-triangularization. Explain your work. (Hint; there is an obvious eigenvalue. Find an eigenvector; use cross- products to complete to an ONB for R³, etc., ) 11. Let Amxn and Bmxn be real matrices related via B = SAT, where S is mxm real orthogonal and T is nxn real orthogonal. Show that A and B have the same singular values and hence the same spectral norms. 12. Consider the system of equations 2x1 + x2 = 0, 0x1 + 0x2= 1, 1 + 2x2 = -1, Let A be the coefficient matrix of this system of equations. Find its SVD. Use that to find the Moore-Penrose inverse of A and thereby find the least squares solution to the system.
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