Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.
-$\left[ \begin{array} { l l } 3 & 2 \ 2 & 6 \end{array} \right]$
A)
$$P = \left[ \begin{array} { r r }
1 & - 2 \
2 & 1
\end{array} \right] , D = \left[ \begin{array} { l l }
8 & 0 \
0 & 3
\end{array} \right]$$
B)
$$P = \left[ \begin{array} { r r }
1 / \sqrt { 5 } & - 2 / \sqrt { 5 } \
- 2 / \sqrt { 5 } & 1 / \sqrt { 5 }
\end{array} \right] , D = \left[ \begin{array} { l l }
2 & 0 \
0 & 7
\end{array} \right]$$
C)
$$P = \left[ \begin{array} { r r }
1 & - 2 \
2 & 1
\end{array} \right] , D = \left[ \begin{array} { l l }
7 & 0 \
0 & 2
\end{array} \right]$$
D)
$$\mathrm { P } = \left[ \begin{array} { c c }
1 / \sqrt { 5 } & - 2 / \sqrt { 5 } \
2 / \sqrt { 5 } & 1 / \sqrt { 5 }
\end{array} \right] , \mathrm { D } = \left[ \begin{array} { l l }
7 & 0 \
0 & 2
\end{array} \right]$$

Question

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.
-$\left[ \begin{array} { l l } 3 & 2 \ 2 & 6 \end{array} \right]$
A)
$$P = \left[ \begin{array} { r r } 
1 & - 2 \
2 & 1
\end{array} \right] , D = \left[ \begin{array} { l l } 
8 & 0 \
0 & 3
\end{array} \right]$$
B)
$$P = \left[ \begin{array} { r r } 
1 / \sqrt { 5 } & - 2 / \sqrt { 5 } \
- 2 / \sqrt { 5 } & 1 / \sqrt { 5 }
\end{array} \right] , D = \left[ \begin{array} { l l } 
2 & 0 \
0 & 7
\end{array} \right]$$
C)
$$P = \left[ \begin{array} { r r } 
1 & - 2 \
2 & 1
\end{array} \right] , D = \left[ \begin{array} { l l } 
7 & 0 \
0 & 2
\end{array} \right]$$
D)
$$\mathrm { P } = \left[ \begin{array} { c c } 
1 / \sqrt { 5 } & - 2 / \sqrt { 5 } \
2 / \sqrt { 5 } & 1 / \sqrt { 5 }
\end{array} \right] , \mathrm { D } = \left[ \begin{array} { l l } 
7 & 0 \
0 & 2
\end{array} \right]$$

Quizplus · Accepted Answer

To orthogonally diagonalize a matrix, we find its eigenvalues and eigenvectors. For the given matrix, the eigenvalues are 7 and 2. The corresponding normalized eigenvectors form the columns of P, and the eigenvalues are the entries of D. Choice D correctly represents the orthogonal matrix P with normalized eigenvectors and the diagonal matrix D with the eigenvalues 7 and 2.

Orthogonally Diagonalize the Matrix, Giving an Orthogonal Matrix P and a Diagonal