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2. Compute the eigenvalues and eigenvectors of each of the following matri- ces. 0 1 -1 1 0 -1 0 2 0 0 1 -2 0 0

2. Compute the eigenvalues and eigenvectors of each of the following matri- ces. 0 1 -1 1 0 -1 0 2 0 0 1 -2 0 0

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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ATH-5342 X FIL PDF Jackson and Ginsburg 2008 p X Ontent/uploads/sites/16915/2018/04/linalg.pdf O 0 1 + 0 Figure 2.1.3. Action of the transformation (2.1.15) on R2, with a dou- ble eigenvalue and one-dimensional eigenspace 2. Compute the eigenvalues and eigenvectors of each of the following matri- ces. BAB = -1 0 2 0 -1 0 W linalg.pdf 2 1 -2 0 A₁ CD A DO 3. Let A E M(n. C). We say A is diagonalizable if and only if there exists an invertible BE M(n. C) such that BAB is diagonal:
Transcribed Image Text:ATH-5342 X FIL PDF Jackson and Ginsburg 2008 p X Ontent/uploads/sites/16915/2018/04/linalg.pdf O 0 1 + 0 Figure 2.1.3. Action of the transformation (2.1.15) on R2, with a dou- ble eigenvalue and one-dimensional eigenspace 2. Compute the eigenvalues and eigenvectors of each of the following matri- ces. BAB = -1 0 2 0 -1 0 W linalg.pdf 2 1 -2 0 A₁ CD A DO 3. Let A E M(n. C). We say A is diagonalizable if and only if there exists an invertible BE M(n. C) such that BAB is diagonal:
Show that this is also given by det (AIA) = n k=0 where do(A1,..., An) = 1, and, for 1 ≤ k ≤n, O k=1 σκ(λι,..., λη) = Σ Aj₁ Ajk 1<<...<jk ≤n The polynomials ok are called the elementary symmetric polynomials. 64 (-1) kok(A₁,..., An)λ¹-k, 8. If A, B E M(n, C), B invertible, and D = B-¹AB, show that, for all KEN, D = B-¹A B. i 9. Let A denote the first matrix in Exercise 2. Diagonalize A and use this to compute A100 2. Eigenvalues, eigenvectors, and generalized eigenvectors
Transcribed Image Text:Show that this is also given by det (AIA) = n k=0 where do(A1,..., An) = 1, and, for 1 ≤ k ≤n, O k=1 σκ(λι,..., λη) = Σ Aj₁ Ajk 1<<...<jk ≤n The polynomials ok are called the elementary symmetric polynomials. 64 (-1) kok(A₁,..., An)λ¹-k, 8. If A, B E M(n, C), B invertible, and D = B-¹AB, show that, for all KEN, D = B-¹A B. i 9. Let A denote the first matrix in Exercise 2. Diagonalize A and use this to compute A100 2. Eigenvalues, eigenvectors, and generalized eigenvectors
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