Hello, could anyone help me out with these two questions.
With question d) I know that because D is an idempotent matrix, then In - D must also be idempotent through:
(In-D)(In-D) = In-2Dn+D = In - D
And also the relation between the linear maps:
S: Rn -> Rn has matrix D
S' : Rn -> Rn had matrix In - D
Then the dot product of S and S' is equal to the multiplication of D and In-D.
But that's as far as I can get with d)
And with e) the only thing I have proved so far is the E is idempotent due to when it's squared it returns E but I don't know what to do next.
Hint for d: $S \circ S' = D \circ (I_n - D) = D \circ I_n - D \circ D$.
Hint for e: For every $n$, $T^{n} = T \circ T^{n-1}$. For example, $$ T^3 = T \circ T^2 = T \circ T = T. $$