I want to find the spectral decomposition of the of the matrix $B$ given the following information:
A =\begin{pmatrix}
2 & 1\\
1 & 2\\
\end{pmatrix}
with $c \gt 0$ and $B = cA$.
I found that for $A$ the eigenvalues are as follows: $\lambda_1 = 3, \lambda_2 = 1$. Furthermore, after normalizing, the corresponding eigenvectors are
$$\bf{v_1} = \frac{1}{\sqrt2}
\begin{pmatrix}
1\\
1\\
\end{pmatrix}$$
and
$$\bf{v_2}= \frac{1}{\sqrt2}
\begin{pmatrix}
1\\
-1\\
\end{pmatrix}$$
Best Answer
You can use the eigenvectors you found (or just common sense) to deduce that the spectral decomposition of $A$ is $$ A=3\,\begin{bmatrix}1/2&1/2\\1/2&1/2\end{bmatrix}+1\,\begin{bmatrix}1/2&-1/2\\-1/2&1/2\end{bmatrix}. $$ So $$ B=3c\,\begin{bmatrix}1/2&1/2\\1/2&1/2\end{bmatrix}+c\,\begin{bmatrix}1/2&-1/2\\-1/2&1/2\end{bmatrix}. $$