Exercise 4.1.26

$A=\left[\begin{array}{ccc}\hfill 2& \hfill 2& \hfill -1\\ \hfill -1& \hfill 2& \hfill 2\\ \hfill 2& \hfill -1& \hfill 2\end{array}\right],\begin{array}{ccc}\hfill & \text{ This example shows a matrix with perpendicular columns. }\hfill & \hfill \\ \hfill & A^{T}A=9I\text{ is diagonal: }{\left(A^{T}A\right)}_{\mathit{ij}}=(\text{ column }i\text{ of }A)\cdot (\text{ column }j\text{ of }A).\hfill \\ \hfill & \text{ When the columns are unit vectors, then }{A}^{T}A=I.\hfill \end{array}$