Boyd's 'Convex Optimization' states that ellipsoids have the form,
$(x-x_c)^TP^{-1}(x-x_c)\leq 1 $; Equation (1)
where P is a symmetric and positive semidefinite matrix and $x_c$ is the centre of the ellipse. This part is clear enough.
It also states that the ellipsoid can also be written in the form,
$ x_c + Au \quad ||u||_2\leq1$; Equation (2)
where A is square and nonsingular.
However the book then states that by taking $A=P^{1/2}$, Equation(2) can be described in the form of Equation(1). How did they come to this conclusion?
\begin{eqnarray} \{x | (x-x_c)^T P^{-1} (x-x_c) \le 1 \} &=& \{x_c\}+ \{x | x^TP^{-1} x \le 1 \} \\ &=& \{x_c\}+ \{x |\| P^{- {1 \over 2}} x\| \le 1 \} \\ &=& \{x_c\}+ \{P^{{1 \over 2}}x | \ \| x\| \le 1 \} \\ \end{eqnarray}