Diagonalization: How to show that A exists $S^2 = D$ given that D is a nonegative diagonal marix

Question

(a) Show that if D is a diagonal matrix with nonnegative entries on the main diagonal, then there is a matrix S such that $S^2 =D$ SOLVED

(b) Show that if A is a diagonalizable matrix with nonnegative eigenvalues, then there is a matrix S such that $S^2 = A$.

I'm completely confused i know that $A = PDP^{-1}$

Question A $$D=\begin{bmatrix} A&0&0 \\ 0&B&0\\ 0&0&C \end{bmatrix}$$

then if $S^2 = D$ then....

$$S=\begin{bmatrix} \sqrt{A}&0&0 \\ 0&\sqrt{B}&0\\ 0&0&\sqrt{C} \end{bmatrix}$$

Answer 1

For part (b), you can apply part (a). For suppose $A=P D_A P^{-1}$, and consider $S$ of the form $P D_S P^{-1}$. Then $S^2=P D_S^2 P^{-1}$.(Check this; it is an important property of similarity transformations.) So you just need a diagonal matrix $D_S$ such that $D_S^2=D_A$. But you've just proven that there is one.