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fixed walled and two masses connected by springs

When you write newton's 2nd for each mass you can assume a sinusoid solution for the position as a function of time and plugging back into your equations you can solve for the two fundamental frequencies of the system. My question is this, If i write both equations in a matrix form

[tex]

\frac{d^2 \, \vec{x}}{dt^2} = A \, \vec{x}

[/tex]

with A being a matrix of combinations of spring constants divided by appropriate masses, do the eigenvalues of the matrix correspond to the squares of the natural angular frequencies? If so, why physically is this true? Additionally, what do the eigenvectors mean? What is the physical connection there?

-John