Answer the following questions about a real symmetric matrix, $\boldsymbol { A }$ : $$A = \left( \begin{array} { l l l }
0 & 1 & 2 \\
1 & 0 & 2 \\
2 & 2 & 3
\end{array} \right)$$ I. Find all the different eigenvalues of matrix $\boldsymbol { A } , \lambda _ { 1 } , \cdots , \lambda _ { r } \left( \lambda _ { 1 } < \cdots < \lambda _ { r } \right)$. II. Find all the eigenspaces $W \left( \lambda _ { 1 } \right) , \cdots , W \left( \lambda _ { r } \right)$ corresponding to $\lambda _ { 1 } , \cdots , \lambda _ { r }$, respectively. III. Find an orthonormal basis, $\boldsymbol { b } _ { 1 } , \boldsymbol { b } _ { 2 } , \boldsymbol { b } _ { 3 }$, which belongs to either of $W \left( \lambda _ { 1 } \right) , \cdots , W \left( \lambda _ { r } \right)$, obtained in Question II. IV. Find the spectral decomposition of $A$ : $$A = \sum _ { i = 1 } ^ { r } \lambda _ { i } P _ { i }$$ where $\boldsymbol { P } _ { i }$ is the projection matrix onto $W \left( \lambda _ { i } \right)$. V. Find $A ^ { n }$, where $n$ is any positive integer.
\section*{Problem 2}
Answer the following questions about a real symmetric matrix, $\boldsymbol { A }$ :
$$A = \left( \begin{array} { l l l }
0 & 1 & 2 \\
1 & 0 & 2 \\
2 & 2 & 3
\end{array} \right)$$
I. Find all the different eigenvalues of matrix $\boldsymbol { A } , \lambda _ { 1 } , \cdots , \lambda _ { r } \left( \lambda _ { 1 } < \cdots < \lambda _ { r } \right)$.
II. Find all the eigenspaces $W \left( \lambda _ { 1 } \right) , \cdots , W \left( \lambda _ { r } \right)$ corresponding to $\lambda _ { 1 } , \cdots , \lambda _ { r }$, respectively.
III. Find an orthonormal basis, $\boldsymbol { b } _ { 1 } , \boldsymbol { b } _ { 2 } , \boldsymbol { b } _ { 3 }$, which belongs to either of $W \left( \lambda _ { 1 } \right) , \cdots , W \left( \lambda _ { r } \right)$, obtained in Question II.
IV. Find the spectral decomposition of $A$ :
$$A = \sum _ { i = 1 } ^ { r } \lambda _ { i } P _ { i }$$
where $\boldsymbol { P } _ { i }$ is the projection matrix onto $W \left( \lambda _ { i } \right)$.
V. Find $A ^ { n }$, where $n$ is any positive integer.