Exercise 22

Square matrices are similar if they represent the same transformation, but each with respect to the same ending as starting basis. That is, ${\rm Rep}_{B_{1},B_{1}}(t)$ is similar to ${\rm Rep}_{B_{2},B_{2}}(t)$.

• Give a definition of matrix similarity like that of Definition 2.4.

• Prove that similar matrices are matrix equivalent.

• Show that similarity is an equivalence relation.

• Show that if $T$ is similar to $\hat{T}$ then $T^{2}$ is similar to $\hat{T}^{2}$, the cubes are similar, etc.

• Prove that there are matrix equivalent matrices that are not similar.