Texts: #18 please will rate
EXERCISES The following list of matrices and their respective characteristics:
A = 13
B = 1 - 2
C = -14 - 2
D = 3 2
E = -1 - 1 - 1 1
HE
(t1)
P = r + 2r - 2
p = 1 + 1t + 5 - 15
In Exercises 1-11, find a basis for the eigenspace E for the given matrix and the value of λ. Determine the algebraic and geometric multiplicities of λ.
1. A, λ = 2
2. A, λ = 1
3. B, λ = 2
4. C, x = 1
5. C, x = -1
6. D, λ = 1
7. E, x = -1
8. E, a = 5
9. E, λ = 15
10. F, x = -2
11. F, x = 2
In Exercises 12-17, find the eigenvalues and the eigenvectors for the given matrix. Is the matrix defective?
12. [1 2; 0 -1], not provided
13. [2 0; 1 2], not provided
14. [1 2 1; 0 1 2; 0 0 1]
15. [2 0 3; 0 2 0]
16. [-1 6 -2; 0 5 -6; -1 2 -1]
17. [3 -1 -1; 0 5 -6; -1 2 0]
In Exercise 21, any matrix P is called idempotent if P^2 = P. Show that if P is an invertible idempotent matrix, then P = I.
22. Let P be an idempotent matrix. Show that the only eigenvalues of P are 0 and 1. Hint: Suppose that Px = λx.
23. Let u be a vector in R such that ||u|| = 1. Show that the n x n matrix P = uu^T is an idempotent matrix. Hint: Use the associative properties of matrix multiplication.
24. Verify that if Q is idempotent, then so is -Q. Also, verify that 1 - 2Q = 1 - 2.
25. Suppose that u and v are vectors in R such that ||u|| = 1, ||v|| = 1, and u · v = 0. Show that P = uu^T + vv^T is idempotent.
26. Show that any nonzero vector of the form au + b is an eigenvector corresponding to λ = 1 for the matrix P in Exercise 25.
14. [1 2 1; 0 1 2; 0 0 1]
15. [2 0 3; 0 2 0]
16. [-1 6 -2; 0 5 -6; -1 2 -1]
17. [3 -1 -1; 0 5 -6; -1 2 0]
18. If a vector x is a linear combination of eigenvectors of a matrix A, then it is easy to calculate the
