I am asked to construct a $4 \times 4$ symmetric matrix, with given eigenvalues and eigenvectors. I understand how to actually get $A$ as a product of $P^T, D$ and $P$, when $D$ is the diagonal matrix, and $P$ is a matrix with the eigenvectors as columns.
The problem is that there is only three given eigenvectors, along with three eigenvalues (one is repeated), so my question is, how do you construct a $4 \times 4$ matrix with three eigenvectors?
For more information here is the actual question:
Let $A$ be a symmetric $4 \times 4$ matrix with real entries whose eigenvalues are $−1$ and $2$. If $(1, 0, 0, −1)$, $(0, 1, 1, 0)$ is a basis for the eigenspace of eigenvalue $-1$ and $(1, 0, 0, 1)$ is an eigenvector of $A$ with eigenvalue $2$, find the matrix $A$.