Finding other eigenvector and matrix $A$ given eigenvalues I want to find a symmetric matrix $A$, whose eigenvalues are $4$ and $-1$. One of the eigenvectors corresponding to the eigenvalue $4$ is $(2,3)$. I want to find an eigenvector corresponding to the eigenvalue $-1$ and then find the matrix $A$.

Recall that the eigenspaces of a symmetric matrix are mutually orthogonal. Thus, all eigenvectors with eigenvalue $-1$ are orthogonal to all eigenvectors with eigenvalue $4$. Can you come up with a nonzero vector that’s orthogonal to $(2,3)$? Once you’ve done that, you have a basis of $\mathbb R^2$ that consists of eigenvectors of $A$, therefore $A$ is diagonalizable. Can you construct $A$ from its diagonalization?
