What Is the Eigenvalue Problem and How Do You Solve It?

Photo: Rhett Allain. Angular momentum as an example of the eigenvalue problem

This won’t make sense (yet), but the eigenvalue problem looks like this:

This equation says that a matrix (A) operating on a vector (r) is equal to a scalar (λ) multiplied by the same vector. I actually remember seeing this for the first time and just thinking — “hey, wouldn’t A just be equal to λ?”. Alas, no. A is a matrix and λ is a scalar. They can’t be equal.

Angular Velocity and Eigenvalues

Perhaps a better way to understand this eigenvalue problem is to look at a real situation dealing with angular velocity and angular momentum. The first time you see angular momentum (in your intro-physics course), it looks like this:

Here, L is the angular momentum, ω is the angular velocity and I is the moment of inertia. The moment of inertia is a way to describe the “rotational mass” — or the object’s resistance to changes in rotational motion (just like the mass (m) is a resistance to changes in velocity). I is defined as:

But this definition of the moment of inertia is only true for the case of an object rotating about a fixed axis. If the object is free to rotate in any direction then I is no longer a scalar value but instead a tensor (hint: we call it the moment of inertia tensor). It looks like this: