D and E are nxn matrices, E is invertible, DE = ED, and u is an eigenvector for D corresponding to x=5.
a. Show that Eu is also an eigenvector for D corresponding to x=5.
b. Show that u is an eigenvector for D^2.
c. Show that u is an eigenvector for
D^2 - 3D.
See the attached file.
a) We know:
Du=5u (u is eigenvector for 5) and DE=ED(*). Let's multiply the first equality by E to the left.
Then, EDu=5Eu so DEu=5Eu because of ...
The solution provides a proof regarding eigenvectors and matrices.