When to use eigenvector method

[imaloonru]

I'm a physics major. As such, I have come across several situations in my studies that require the use of eigenvectors and eigenvalues. Whenever I have to use this method, I've been told to. I do not have a complete understanding of eigenvectors and values and am wondering how you would spot a situation where you would need to use them.

For example, if I wanted to know when or where the rate of change of something was 0, I would take a derivative, set it equal to zero, then solve for some variable. What sort of situation would I look for (in general) that would make me say "Hey! I need to find some eigenvectors here."?

Thanks.

 

[Redbelly98]

Welcome to PF.

In general, if you have an equation of the form

Matrix x Vector = Scalar x Vector​

or

Operator x Function = Scalar x Function​

The "Scalar" is an eigenvalue that you must find.

 

[HallsofIvy]

Also you can "simplify" linear transformations, writing them as matrices in either diagonal or Jordan normal form, with eigenvalues on the diagonal, by using the eigenvectors (or if there is not a complete set of eigenvectors, the "generalized" eigenvectors) as basis for the vector space.

 

[imaloonru]

Thanks guys. This helps.

 

[blue_raver22]

I can provide some insight on when to use the eigenvector method. Eigenvectors and eigenvalues are mathematical tools that are commonly used in linear algebra, but they have applications in various fields such as physics, engineering, and computer science.

One situation where you may need to use eigenvectors is when you are dealing with systems that involve transformations or rotations. Eigenvectors represent the directions along which a transformation or rotation does not change, and eigenvalues represent the magnitude of this change. This can be useful in understanding the behavior of physical systems, such as the movement of a pendulum or the rotation of a rigid body.

Another common application of eigenvectors is in data analysis and pattern recognition. In this case, eigenvectors are used to find the principal components of a dataset, which can help identify underlying patterns and relationships between variables.

In general, if you are dealing with matrices or transformations, or trying to analyze patterns in data, it is worth considering the use of eigenvectors. It is also important to note that the eigenvector method is not the only approach to solving problems, so it is always beneficial to have a good understanding of other mathematical techniques and choose the most appropriate one for the specific problem at hand.

I would also recommend further exploring the concept of eigenvectors and eigenvalues to gain a better understanding of their applications and how they can be used in your studies.