Math 54 quiz 7 suppose x is an eigenvector of a

Question

MATH 54 QUIZ 7-

1. Find an invertible matrix P and a matrix C of the formsuch that A =has the form A = PCP^-1.

2. Give an example of a 2 × 2 matrix (with real coefficients) that has no eigenvectors in R².

3. Suppose x is an eigenvector of A corresponding to an eigenvalue λ.

(i) Show that x is an eigenvector of 5I - A. What is the corresponding eigenvalue?

(ii) Show that x is an eigenvector of 5I - 3A + A². What is the corresponding eigenvalue?

4. Give an example of a vector in R4 orthogonal to v = [1, 4, 2, 6]^T.

5. Give an example of three vectors u₁, u₂, u₃ in R² such that u₁ · u₂ = 0 and u₂ · u₃ = 0 but u₁ · u₃ ≠ 0.

6. Let. Compute the distance from y to the line through u and the origin.

7. Does there exist an orthonormal matrix such that |a₁| + |a₂| + · · · + |a₉| > 3? If yes, give an example; if no, give a proof explaining why.