Scalar product of two vectors when only given equations

Question

Let $A$ be a symmetrical $2x2$ matrix. Given the equations

$$A*v_1=4*v_1$$

$$A*v_2=-v_2$$

Determine the scalar product $v_1*v_2$

Any hints?

Answer 1

It can be seen that $v_1$ and $v_2$ are two Eigenvectors corresponding to two different Eigenvalues ($4$ and $-1$) of $A$. Eigenvectors of real symmetric matrices are orthogonal.

Therefore $<v_1,v_2>=0$.

For generalized proof see: Eigenvectors of real symmetric matrices are orthogonal