Chapter 7.3, Problem 28P

We call a matrix $B$ a square root of $A$ if $B^2=A$.

(a) Show that if $D=\text{diag}\left({\lambda }_1,{\lambda }_2,\mathrm{......},{\lambda }_n\right)$, then the matrix

$\sqrt{D}=\text{diag}\left(\sqrt{{\lambda }_1},\sqrt{{\lambda }_2},\mathrm{..........},\sqrt{{\lambda }_n}\right)$

is a square root of $D$.

(b) Show that if $A$ is a non-defective matrix with $S^{-1}AS=D$ for some invertible matrix $S$ and diagonal matrix $D$, then $S\sqrt{D}{S}^{-1}$ is a square root of $A$.

(c) Find a square root for the matrix

$A=\left[\begin{array}{cc}6& -2\\ -3& 7\end{array}\right]$