Problem 8. Supppose A is an invertible nxn matrix and i is an eigenvector of A with associated eigenvalue -2. Convince yourself that i is an eigenvector of the following matrices, and find the associated eigenvalues a The matrix A has an eigenvalue b. The matrox A 'has an eigenvalue C The matrox A + 91, has an eigenvalue Activate ido d. The matrix 2A has an eigenvalue 01:27 17°C AGA 40 ENG Au Ai Xd 2021/08/25 O Type here to search

Problem 8. Supppose A is an invertible nxn matrix and i is an eigenvector of A with associated eigenvalue -2. Convince yourself that i is an eigenvector of the following matrices, and find the associated eigenvalues a The matrix A has an eigenvalue b. The matrox A 'has an eigenvalue C The matrox A + 91, has an eigenvalue Activate ido d. The matrix 2A has an eigenvalue 01:27 17°C AGA 40 ENG Au Ai Xd 2021/08/25 O Type here to search

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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We have the following matrix where each column is identical: a а а А — b 1. Since this matrix is not full rank, it is singular and thus 1 = 0 is an eigenvalue. Find two linearly independent eigenvectors of A corresponding to 1 = 0. You may use SymPy or other libraries to check your work, but please show the steps of your would-be computations. 2. 1 = 0 is also an eigenvalue of A". Find two linearly independent eigenvectors of AT corresponding to 1 = 0. 3. A and A' have three eigenvalues each, but 1 = 0 makes up two of the three since it corresponds to two linearly independent eigenvectors for both matrices. Find the third eigenvalue of A and A' by looking at the trace of A. 4. Show that (a, b, c) is the corresponding eigenvector of A and that (1, 1, 1) is the corresponding eigenvector of A' .
If ₁ and U₂ = A are eigenvectors of a matrix A corresponding to the eigenvalues A₁ = -1 and A₂ = -3, respectively, E then A(1 + ₂) = and A(-30₁)
1 Mark as TRUE or FALSE. If they are true give an explanation, if they are false, give a coun- terexample. You really need to understand the reasons behind your statements. (a) Every square matrix has an eigenvalue over the real numbers. (b) Let A be an n x n matrix. A vector v can be an eigenvector with respect to two different eigenvalues. (c) If the vector v is an eingenvector for the eigenvalue = ; show that lim, m A"v = 0, where 0 is the zero vector 0,0, .. ., 0]. (d) Let A be an n x n invertible matrix. A vector v can be an eigenvector with respect to an eigenvalue c, then v is an eigenvector for A- with respect to the eigenvalue c 4 (e) 5 vectors in R are always linearly dependent. (f) 4 vectors in R are always linearly independent. 2. Decide whether the vectors (1, 2,0, -1], [2, 6, -3, -3], [3, 10, -6,-5] are linearly independent. What is the dimension of the span they generate? 3. Find a basis among the vectors (1, 2, 3], [2,6, 10], [0, –3, -6], [-1, -3, -5] (which are surely…
If matrix A is square of size n x n with a rank equal to 2 and n > 2. The rank of a matrix is equal to the number of non-zero eigenvalues. Can you write A as the product of two matrices B E Rnxq and C E RPX" where we require that p < n and q < n? You may assume that A is diagonalizable. Show how to construct a matrix B and C such that A = BC.
Supppose A is an invertible ŉ x n matrix and is an eigenvector of A with associated eigenvalue -6. Convince yourself that is an eigenvector of the following matrices, and find the associated eigenvalues. a. The matrix A9 has an eigenvalue b. The matrix A-¹ has an eigenvalue c. The matrix A + 7In has an eigenvalue d. The matrix 9A has an eigenvalue
2. Find the eigenvalues and normalized eigenvectors of the following matrix. Note: This one looks like it will lead to a cubic, but multiply it out to get an equation that can be simply solved. 7 1 2 1 7 2 2 2 1
6. Consider the matrix A = Compute the eigenvalues of A, and the corresponding eigenspaces.
A legitimate eigenvector of a 10 x 10 matrix could be [0] 0 0 0 0 0 0 0 [0] O [1 2 3 4 5 6 7 8 9 10 T O 2 3 4 5 6 7 SALLE DE za katere WE None of the first 3 choices can be correct. All of the first 3 choices can be correct
Mark as TRUE or FALSE. You will need to give an explanation. (a) If v1, v2 ..., Vn are linearly independent, then v1, V2 are linearly independent. (b) If v and w are two eigenvectors for the same eigenvalue c, then the sum v + w is also an eigenvector for the eigenvalue c. (c) Any 2 x 2 matrix has a real eigenvalue. (d) If v is an eigenvector for a matrix A and a matrix B then v is an eigenvector for A+ B.
9. Determine which of the following vectors 1 -1 is an eigenvector of the matrix 1 1 1 1 -1 1 -1 W = -1 -1 1 -1 -1 1 and find the corresponding eigenvalue.