Let $\pi$ be the plane spanned up by vectors $v_1 = (1, 1, 1)$ and $v_2 = (2, 1, 3)$. Consider linear transformation that stretches the plane in the direction of $v_1$ by factor $2$ and in the direction of $v_2$ by factor $3$. Find all eigenvectors and eigenvalues for this transformation. In addition, find an explicit form of transformation matrix.
The problem fix the eigenvalue $\lambda_1=2$ with eigenvector $v_1$ and the eigenvalue $\lambda_2=3$ with eigenvector $v_2$.
Without other conditions the other eigenvalue $\lambda_3$ can have any real value (note that it cannot have a complex value. Whay?).