Formulating an Eigenvector Equation

• 2.718281828459
In summary, The conversation discusses finding the normal mode frequencies for a double pendulum and the use of matrices in the derivation process. The speaker suggests finding the eigenvalues of M^-1K to solve for the desired frequency, but also mentions another method involving finding the determinant of K-w^2M. The speaker also offers advice to check for mistakes in the matrices used.
2.718281828459
Hello. I am working on a project with a double pendulum and I am currently looking for the normal mode frequencies. I don't think that's too important to answer my question, but in the derivation I hit a point that look like this:$(K-M\omega^{2})\alpha=0.$ Here, K and M are 2x2 square matrices. I want to solve for the eigenvalue here, but this doesn't follow the form that I normally have with eigenvector equations. If I rearrange this, I get $K\alpha=M\omega^{2}\alpha$ which has a matrix on each side. Naturally I thought to multiply by the inverse of M on both sides to get $M^{-1}K\alpha=\omega^{2}\alpha.$ However, this didn't give the correct result. Why not? How should it be done?

You must have K or M wrong, or both.
Check again, or submit the original problem and your approach here.

Finding the eigenvalues of ##M^{-1}K## works. You must have made a mistake somewhere.

But you don't need to do that. Just find the determinant of ##K-\omega^2M##, and find the values of ##\omega## that make it zero.

$$\left|\begin{matrix} k_{11}-\omega^2m_{11} & k_{12}-\omega^2m_{12} \\ k_{21}-\omega^2m_{21} & k_{22}-\omega^2m_{22} \end{matrix} \right| = (k_{11}-\omega^2m_{11})(k_{22}-\omega^2m_{22}) - (k_{12}-\omega^2m_{12})(k_{21}-\omega^2m_{21})$$

1. What is an eigenvector equation?

An eigenvector equation is a mathematical expression that represents the relationship between an eigenvector and its corresponding eigenvalue. It is commonly used in linear algebra to solve problems involving matrices.

2. How do you formulate an eigenvector equation?

To formulate an eigenvector equation, you need to first find the eigenvalues of the given matrix. Then, for each eigenvalue, you need to find the corresponding eigenvector by solving the equation (A - λI)x = 0, where A is the original matrix, λ is the eigenvalue, and x is the eigenvector.

3. Why is the eigenvector equation important?

The eigenvector equation is important because it allows us to find the eigenvectors of a matrix, which are used in many applications such as data analysis, image processing, and quantum mechanics. It also helps us understand the behavior of a system represented by a matrix.

4. How is the eigenvector equation used in data analysis?

In data analysis, the eigenvector equation is used to find the principal components of a dataset. These principal components are the eigenvectors of the covariance matrix of the data and can provide insights into the underlying structure and patterns in the data.

5. Can the eigenvector equation be solved for any matrix?

No, the eigenvector equation can only be solved for square matrices. Additionally, not all square matrices have eigenvectors. Only matrices with distinct eigenvalues have unique eigenvectors.

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