**Problem 1.** Let \( A = \begin{pmatrix}-8 & 2 & 2 \\2 & -5 & 4 \\2 & 4 & -5\end{pmatrix} \).1. Compute the characteristic polynomial of the matrix \( A \).2. Find the eigenvalues of \( A \) and their multiplicities.Define a dynamical system with \( b_0 = \begin{pmatrix}1 \\2 \\3\end{pmatrix} \) and \( b_{n+1} = A b_n \), for \( n = 1, 2, 3, \ldots \), with \( A \) as in Problem 1. Find a formula for \( b_n \) for each \( n \).

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Answered: Problem 1. Let A = 2 2 2 -5 4 4 -5 (1)…

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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A 2x2 upper triangular matrix A has eigenvalues a and - 5 (with a ≠ -5). An eigenvector corresponding to the eigenvalue a is (1,0). Which of the following could be the matrix A? A. (-5 a/2 0) B. (a 0/2 -5) C. (-5 2/0 a) D. (a -5/0 2) E. (a 2/0 -5)
4. (a) Show that if A is a diagonalizable matrix with non-negative real eigenvalues, then there is a matrix S such that S? = A. (b) Using the general method from (a), find a matrix whose square is 1 0 8 0 4 0 0 0 9 A =
Problem 6 In Exercises 1 - 2 determine the characteristic polynomial and the eigenvalues of the given matrices. -9 -2 -10 §7.2, Exercise 1. A= 3 3 8 2 9.
Find the matrix A of the quadratic form associated with the equation. бx2 - 5ху - бу? + 10 %3D 0 6. -2.5 A = |-2.5 -6 Find the eigenvalues of A. (Enter your answers as a comma-separated list.) 13 13 2 2 Find an orthogonal matrix P such that P'AP is diagonal. (Enter the matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list.) P =
1 Let P = -2 - Find p P. Deduce P, if it exists, from the result of -1 p' P. You can also calculate P 1 using the traditional approach but this will be slower. 2. Let A be a 3 x 3 matrix with the following eigen-pairs: 1 (1, ):(1, 0 ):G -2 ).
2. (1) Compute all eigenvalues and eigenvectors of the Cartan matrix 2 -1 C = -1 2 -1 -1 2 (2) Diagonalize C (i.e. find Q such that Q-'CQ is diagonal). (3) Is C positive definite?
6. Consider the matrix A 0 [21] a (a) For what value(s) of a does A have an eigenvalue of multiplicity 2? What is the value of the eigenvalue in this case? (b) For what value(s) of a does A have -1 and 2 as eigenvalues? (c) For what value(s) of a does A have a complex eigenvalues?
[1 2. (a) Find the eigenvalues and eigenvectors of B = (b) Explain why the matrix A cannot be diagonalized.
7. Let A = -2 -2 2 2 -1 4 -2 4 (a). Find the eigenvalues and corresponding eigenvectors for A. (b). Find a matrix P that orthogonally diagonalizes A.
2 3 5. Let A = 1 2 3 3 (a) corresponding eigenspace. For each eigenvalue of A (which you found in #1(c)), find a basis for the (b) If possible, diagonalize A; if this is not possible, then explain why not. Repeat parts (a) and (b) with the matrix 3 1 2] from #1(a). (c)

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Problem 1. Let A = 2 2 2 -5 4 4 -5 (1) Compute the characteristic polynomial of the matrix A. (2) Find the eigenvalue of A and their multiplicities.