Math Geeks: Advanced Q's on Matrices...

What is the practical application of finding “kernel” of a matrix? Also, what is the practical application to finding “Eigen values”? Is there a simple explanation to how we apply these things in science or technology? - Jinx

I’ve not come across the term “kernel” in relation to a matrix.

But eigenvalues are pretty straightforward.

If your matrix describes a system of equations (simultaneous linear equations, simulataneous differential equations, etc.,) the eigenvalues tell you things about your system.

If it’s a system of simultaneous linear equations, the eigenvalues tell you whether the system is close to being singular (indeterminate, having multiple solutions, etc.)

If it’s a system of simultaneous ODEs, the eigenvalues tell you what the resonant frequencies of the system are, and how quickly the resonances are damped out.

The kernel of a matrix is just another name for the null space. Finding the null space is very important when solving a system of linear equations, because the null space is parallel to the space of solutions and hence the null space together with only one solution will give you all solutions.

Forgot to mention another use for eigenvalues: given a vector field, the eigenvalues of the Hessian (I believe) tell you how the vector field behaves near a stationary point. For example, if the eigenvalues are all positive then the stationary point is a source; all negative implies a sink.

Wow! I was just taught to find these things without any understanding of why we were doing such. That’s what happens when you’re taught math robotically with no value for the practical side. As usual, there is always more and more to know!

Thanks,

  • Jinx

I’m sorry, you can’t be told anything about matrices, you’ll just have to be shown :wink:

Eigenvalues are tremendously valuable. They apply in many Sciences. In college, I was once taking 3 classes in 3 fields that discussed them the same quarter.

In Computer Science, suppose you want to raise a matrix to a great power (which we like to do for various reasons). If you have the Eigenvectors, you can do this a whole lot faster than by repeated squaring of the matrix. But finding Eigenvalues isn’t cheap either…