Determine whether the matrix is orthogonal.
$P = \begin{bmatrix} 1/3 & 2/3 & 2/3 \\ 2/3 & -2/3 & 1/3 \\ 2/3 & 1/3 & -2/3 \end{bmatrix}$
Find $PP^T$.
$\begin{bmatrix} 1/3 & 2/3 & 2/3 \\ 2/3 & -2/3 & 1/3 \\ 2/3 & 1/3 & -2/3 \end{bmatrix} \begin{bmatrix} 1/3 & 2/3 & 2/3 \\ 2/3 & -2/3 & 1/3 \\ 2/3 & 1/3 & -2/3 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
Is the matrix $P$ orthogonal?
$P$ is orthogonal.
Let $P_1 = \begin{bmatrix} 1/3 \\ 2/3 \\ 2/3 \end{bmatrix}$, $P_2 = \begin{bmatrix} 2/3 \\ -2/3 \\ 1/3 \end{bmatrix}$, and $P_3 = \begin{bmatrix} 2/3 \\ 1/3 \\ -2/3 \end{bmatrix}$. If the matrix $P$ is orthogonal, show that the column vectors of the matrix form an orthonormal set. (If the matrix is not orthogonal, enter NOT ORTHOGONAL.)
Find $P_1 \cdot P_2$.
Find $P_1 \cdot P_3$.
Find $P_2 \cdot P_3$.
Find $||P_1||$.
Find $||P_2||$.
Find $||P_3||$.
