Using linear algebra to find resonance frequency and normal oscillations and motion I am stuck part way through the following and not sure how or if finding eigenvalues will help with finding modes of oscillations:
Consider the system of three masses and two ideal elastic bands:
$(m)$---$k$---$(2m)$---$2k$---$(m)$  [$m$ are masses, $k$ is spring constant]
Find the resonance frequencies of oscillation, normal modes of oscillation and describe motion of masses corresponding to modes of oscillations.
My work so far:
$$V=\frac{1}{2}(k9x-y)^{2}+k(y-z)^{2}=\frac{1}{2}k[x^{2}-2xy+3y^{2}-4yz+2z^{2}]$$
$$m\ddot{x}=-\frac{\partial V}{\partial x}=-k(x-y)$$
$$m\ddot{y}=-\frac{\partial V}{\partial y}=-k(-x+3y-2z)$$
$$m\ddot{z}=-\frac{\partial V}{\partial z}=-k(2z-2y)$$
$$\therefore\ \ m\ddot{x}+m\ddot{y}+m\ddot{z}=0$$ 
$$y=\frac{-1}{2}(x+z)$$
$$\therefore -m\omega^{2}x=-\frac{1}{2}k(3x+z)$$ 
$$\therefore -m\omega^{2}y=-k(x+3z)$$ 
Up to here I'm good. But from here, to find the normal modes of oscillation I am not sure where to go. If I define $\lambda=\frac{m\omega^{2}}{k}$(from an example in text) I get:
$$\lambda(X) = \left[ \begin{array}{cc}
\frac{3}{2} & \frac{1}{2} \\
1 & 3 \\ \end{array} \right](X)$$
Would I then use $(A-\lambda I)(X)=0$ to find eigenvalues? If I do so, I get:
$\lambda_{1,2}=\frac{1}{4}(9 \pm \sqrt{17})$. At this point I begin to lose confidence and understanding in my method. Any help or guidance would be appreciated. Also, I apologize if the formatting is off putting, I am still learning the language. 

Without using the condition $m\ddot{x}+m\ddot{y}+m\ddot{z}=0$, just write your 3D system as 
$$ k\left[ \begin {array}{ccc} -1&1&0\\1&-3&2
\\ 0&2&-2\end {array} \right] \left[ \begin {array}{c} x\\y\\z\end {array} \right]=m\left[ \begin {array}{c} \ddot{x}\\\ddot{y}\\\ddot{z}\end {array} \right]$$
Now compute the frequencies as eigenvalues and normal modes of oscillation as eigenvectors.
Notice that the eigenvalue $0$, corresponding to uniform motion of the system as a whole, appears naturally.
