Find the eigenvectors corresponding to an eigenvalue I know how to find the eigenvectors corresponding to an eigenvalue of a matrix $A$: we basically need to find the vectors of the nullspace of $\lambda I - A$, but in my case, I have a matrix $A$ like this:
$$A = \left(\begin{matrix} \cos x & -\sin x \\ \sin x & \cos x\end{matrix}\right) \left(\begin{matrix} 3 & 0 \\ 0 & 7\end{matrix}\right) \left(\begin{matrix} \cos x & \sin x \\ -\sin x & \cos x\end{matrix}\right)$$
and I cannot multiply the individual component matrices.
I have found the eigen values $\lambda = 3$ and $\lambda = 7$ for the matrix $A$, now how do I found the eigen vectors, without multiplying $A$ first?

Eigenvector for eigenvalue 3
Given that
$$
\left( \begin{array}{cc} 3 & 0 \\ 0 & 7 \end{array} \right)
\left( \begin{array}{c} 1 \\ 0 \end{array} \right)
=
3
\left( \begin{array}{c} 1 \\ 0 \end{array} \right),
$$
we can also write
$$
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{cc} 3 & 0 \\ 0 & 7 \end{array} \right)
\left( \begin{array}{c} 1 \\ 0 \end{array} \right)
=
3
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{c} 1 \\ 0 \end{array} \right),
$$
or
$$
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{cc} 3 & 0 \\ 0 & 7 \end{array} \right)
\left( \begin{array}{cc} \cos(x) & \sin(x) \\ -\sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{c} 1 \\ 0 \end{array} \right)
=
3
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{c} 1 \\ 0 \end{array} \right),
$$
or using $A$:
$$
A
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{c} 1 \\ 0 \end{array} \right)
=
3
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{c} 1 \\ 0 \end{array} \right),
$$
Meaning the the eigenvector for the eigenvalue $3$ is given by
$$
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{c} 1 \\ 0 \end{array} \right)
=
\left( \begin{array}{c} \cos(x) \\ \sin(x) \end{array} \right)
$$
Eigenvector for eigenvalue 7Given that

$$
\left( \begin{array}{cc} 3 & 0 \\ 0 & 7 \end{array} \right)
\left( \begin{array}{c} 0 \\ 1 \end{array} \right)
=
7
\left( \begin{array}{c} 0 \\ 1 \end{array} \right),
$$
we can also write
$$
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{cc} 3 & 0 \\ 0 & 7 \end{array} \right)
\left( \begin{array}{c} 0 \\ 1 \end{array} \right)
=
7
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{c} 0 \\ 1 \end{array} \right),
$$
or
$$
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{cc} 3 & 0 \\ 0 & 7 \end{array} \right)
\left( \begin{array}{cc} \cos(x) & \sin(x) \\ -\sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{c} 0 \\ 1 \end{array} \right)
=
7
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{c} 0 \\ 1 \end{array} \right),
$$
or using $A$:
$$
A
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{c} 0 \\ 1 \end{array} \right)
=
3
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{c} 0 \\ 1 \end{array} \right),
$$
Meaning the the eigenvector for the eigenvalue $7$ is given by
$$
\left( \begin{array}{cc} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x)
    \end{array} \right)
\left( \begin{array}{c} 0 \\ 1 \end{array} \right)
=
\left( \begin{array}{c} -\sin(x) \\ \cos(x) \end{array} \right)
$$
