How do I find all the eigenvectors of a matrix $A$ that has the following properties?

Question

a) $A$ is a symmetric $4\times 4$ matrix.

b) The rank of the matrix is $2$.

c) Vectors $ \langle1,2,0,1\rangle$ and $ \langle0,1,1,0\rangle$ are eigen vectors with eigenvalue $2$.

Answer 1

Hints:

• For a matrix $A$ of order $n$, $0$ is eigenvalue of order $\geq n - rank(A)$
• All eigenvectors of $A$ are independent

Answer 2

Since the rank of $A$ is $2$ so by the rank-nullity theorem the dimension of the kernel of $A$ is $2$ so $0$ is an eigenvalue with multiplicity $2$ and since $A$ is symmetric then it's diagonalizable so there is an orthogonal basis of eigenvectors of $\Bbb R^4$ hence denote the given vectors $e_1,e_2$ and take two linearly independent vectors $e_3,e_4$ in the subspace $(\operatorname{span}(e_1,e_2))^\perp$.