Define the orthogonal projection in the spectral decomposition for the $2 \times 2$ matrix $ \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$.
Answer:
I can find the orthogonal projection of the above matrix.
Also I can find the spectral decomposition of the above matrix.
But I do not understand the question's meaning.
What actually it means by the word $ \text{define the orthogonal projection in the spectral decomposition}$ ?
Help me
I presume the question requires to find orthogonal projections $P_{\lambda}$ in the spectral decomposition $$\fbox{$A=\lambda_1P_{\lambda_1}+\cdots+\lambda_nP_{\lambda_n}$} $$ where $P_{\lambda}$ denotes the orthogonal projection onto eigenspace $V_{\lambda}$.
$\text{( I am giving details for the ones who do not know )}$
The eigenvalues and corresponding eigenvectors are $\lambda=3\:,\:\lambda=1$ with $\displaystyle V_{\lambda_3}=\begin{bmatrix}1\\1\end{bmatrix}$ and $\displaystyle V_{\lambda_1}=\begin{bmatrix}1\\-1\end{bmatrix}$. The projection matrix onto $V_{\lambda}$ is given by $$\fbox{$P=V_{\lambda}(V_{\lambda}^TV_{\lambda})^{-1}V_{\lambda}^T$} $$ We have $$P_{\lambda_3}=\begin{bmatrix}\dfrac12 &\dfrac12\\\dfrac12 &\dfrac12\end{bmatrix}\:\text{ and}\: P_{\lambda_1}=\begin{bmatrix}\dfrac12 &\dfrac{-1}{2}\\\dfrac{-1}{2}&\dfrac12\end{bmatrix}$$
The spectral decomposition of $A$ is $$A=3\cdot \begin{bmatrix}\dfrac12 &\dfrac12\\\dfrac12 &\dfrac12\end{bmatrix}+1\cdot\begin{bmatrix}\dfrac12 &\dfrac{-1}{2}\\\dfrac{-1}{2}&\dfrac12\end{bmatrix} $$