2
$\begingroup$

Why is this expression: $\begin{pmatrix} \frac{k+mg}{l} & -k\\ -k & \frac{k+mg}{l} \end{pmatrix} \begin{pmatrix} \rho_1\\\rho_2 \end{pmatrix}=\omega^2\begin{pmatrix} m & 0\\ 0 & m \end{pmatrix} \begin{pmatrix} \rho_1\\\rho_2 \end{pmatrix}$

a matricial form for equation for eigenvalues and eigenvectors? I was told that the generical expression of the equation for eigenvalues and eigenvectors is $A\bf{x}=\lambda \bf{x}$... How can I obtain eigenvalues and eigenvectors from the first expression that I have written? Thank you!

1 Answers 1

5

If $A$ is the big matrix on the left and $x = (\rho_1, \rho_2)^T$, then the equation you have written can be rewritten as $ Ax = \omega^2 m I x$ where $I$ is the identity matrix. But this can thus be written as $Ax = \omega^2 m x$, so this is just the eigenvalue-eigenvector equation with eigenvalue $\lambda = \omega^2 m$.

  • 1
    The "norm" of a vector depends on the choice of norm. It sounds like, here, the norm of a vector is given by $ \|(x_1, x_2)\|^2 = m(x_1^2 + x_2^2) $2012-12-18