### Chapter 5 continued

- Execute

D:=Matrix([[c,0],[0,c]]);

Part A: Execute the Eigenvectors command.

Part B: How does this transformation act on R^{2}?

Part C: Use Part B to explain your output in Part A.

- Execute

Sh:=Matrix([[1,k],[0,1]]);

Part A: Execute the Eigenvectors command.

Part B: How does Sh act as a transformation on R^{2} for
positive k values?

Part C: Use Part B to explain your output in Part A.

Part D: Is A diagonalizable? Why or why not?

Part E: If k=0, Maple's response to Part A and your response to Part D are
both incorrect - what should the responses be?

- Execute

R:=Matrix([[cos(2*theta), sin(2*theta)],[sin(2*theta), -cos(2*theta)]]);

Notice the difference between this matrix and a rotation matrix.

Part A: Apply the Eigenvectors(R); and Eigenvalues(R); commands.

Part B:
When theta=0, what geometric transformation is R?
What are the eigenvectors for each eigenvalue in this case?

Part C: When theta=Pi/4, what geometric transformation is R?
What are the eigenvectors for each eigenvalue in this case?

Part D: What is the relationship between theta and the geometric
transformation R(theta)?

Part E: What are the eigenvectors for each eigenvalue for a general
theta (hint: the reasoning and trigonometry
is similar to what we used for the projection matrix): resolve with the
Maple Eigenvectors output.

Part F: Is R(theta) diagonalizable?