¹The book also calls the \(S\) matrix as the shape matrix, so I better show this as well, which is defined as \(S=M^{-\frac{1}{2}}P\), hence

\begin{align*} P & =\begin{bmatrix} \mathbf{v}_{1} & \mathbf{v}_{2}\end{bmatrix} \\ & =\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & -1\\ 1 & 1 \end{bmatrix} \end{align*}

\begin{align*} S & =\begin{bmatrix} 2.\,\allowbreak 5 & 0\\ 0 & 2.\,\allowbreak 5 \end{bmatrix} ^{-\frac{1}{2}}\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & -1\\ 1 & 1 \end{bmatrix} \\ & =\frac{1}{\sqrt{2}}\begin{bmatrix} 2.\,\allowbreak 5^{-\frac{1}{2}} & 0\\ 0 & 2.\,\allowbreak 5^{-\frac{1}{2}}\end{bmatrix}\begin{bmatrix} 1 & -1\\ 1 & 1 \end{bmatrix} \\ & =\sqrt{2}\begin{bmatrix} 0.316\,23 & -0.316\,23\\ 0.316\,23 & 0.316\,23 \end{bmatrix} \\ & =\fbox{$\begin{bmatrix} 0.447\,22 & -0.447\,22\\ 0.447\,22 & 0.447\,22 \end{bmatrix} $} \end{align*}