I am currently going through some of the exercises in Linear Algebra by Hoffman and Kunze, 2nd edition, and I have come across a question that I don't know how to solve. This is Exercise 5 of Section 6.8.
Suppose $T$ is the diagonalizable linear operator on $\mathbb R^3$: $$ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}. $$ Now the part I don't understand is the following:
Use the Lagrange polynomials to write the matrix $A$ in the form
$$A=E_1+2E_2,\:\: E_1+E_2=I,\:\: E_1E_2=0.$$
I have the equation for $$p_j(x)=\frac{x-c_i}{c_j-c_i},$$ but I'm not sure how to apply it to a $3 \times 3$ matrix. Any help would be greatly appreciated. Thanks!