I am having trouble finding the symmetric matrix $A$ given eigenvalues $1$ and $4$ and eigenvector $(1, 1)$ corresponding to eigenvalue $1$.
I feel like I'd have to use the equation $A=PD(P^{-1})$, but I'm having trouble finding the matrix $P$ if I can't find the second eigenvector.
Any help is appreciated, thanks!
Best Answer
To offer a slightly different perspective, due to the Spectral Theorem you have $$ A=P_1+4P_2,$$ where $P_1$ is the orthogonal projection onto the span of $(1,1)^t$, and $P_2$ is orthogonal to $P_1$; that is, $P_2=I-P_1$. Thus $$ P_1=\tfrac12\,\begin{bmatrix} 1&1\end{bmatrix}\begin{bmatrix} 1\\1\end{bmatrix}=\begin{bmatrix}1/2&1/2\\1/2&1/2\end{bmatrix}. $$ And then $$ A=\begin{bmatrix}1/2&1/2\\1/2&1/2\end{bmatrix}+4\begin{bmatrix}1/2&-1/2\\-1/2&1/2\end{bmatrix}=\begin{bmatrix}5/2&-3/2\\-3/2&5/2 \end{bmatrix} $$