(d) Prove that if A does not have any real eigenvalues, then A is similar to a matrix of the form AQ where Q is an orthogonal matrix and A > 0.
(d) Prove that if A does not have any real eigenvalues, then A is similar to a matrix of the form AQ where Q is an orthogonal matrix and A > 0.
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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solve question (d) asap within 15 mins with explanation
![Problem 1. (Classifying non-diagonalizable 2 x 2 matrices.) Let A € R²x2 be a 2 × 2 matrix.
(a) Suppose that A has eigenvalue 0 but is not diagonalizable. Prove that im(A) = Eo, and
conclude from this that A2 = 0.
(b) Let A ER and suppose that A has eigenvalue A but is not diagonalizable. Prove that we have
(A – Al2)? = 0, and deduce from this that Au – AT e E, for every i e R².
[Hint: apply part (a) to the matrix A – AI2.]
(c) Prove that if A has eigenvalue A but is not diagonalizable, then A is similar to
[Hint: consider the basis B = (A® – Aū, T) where ī ¢ E,.]
(d) Prove that if A does not have any real eigenvalues, then A is similar to a matrix of the form
AQ where Q is an orthogonal matrix and A > 0.
'We work over R throughout this problem. So“eigenvalue" means real eigenvalue, "diagonalizable" means diag-
onalizable over IR, and "similar" means similar over R.
2Recall that for each A E R, Ex = {v € R? : Au = Au}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98149bc7-7eca-4a32-ab0c-b62017e09608%2F62c2af56-760a-43c5-a6df-90ae017e846b%2Fzkt0e85_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1. (Classifying non-diagonalizable 2 x 2 matrices.) Let A € R²x2 be a 2 × 2 matrix.
(a) Suppose that A has eigenvalue 0 but is not diagonalizable. Prove that im(A) = Eo, and
conclude from this that A2 = 0.
(b) Let A ER and suppose that A has eigenvalue A but is not diagonalizable. Prove that we have
(A – Al2)? = 0, and deduce from this that Au – AT e E, for every i e R².
[Hint: apply part (a) to the matrix A – AI2.]
(c) Prove that if A has eigenvalue A but is not diagonalizable, then A is similar to
[Hint: consider the basis B = (A® – Aū, T) where ī ¢ E,.]
(d) Prove that if A does not have any real eigenvalues, then A is similar to a matrix of the form
AQ where Q is an orthogonal matrix and A > 0.
'We work over R throughout this problem. So“eigenvalue" means real eigenvalue, "diagonalizable" means diag-
onalizable over IR, and "similar" means similar over R.
2Recall that for each A E R, Ex = {v € R? : Au = Au}.
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