Is basis change ever useful in practical linear algebra?

Question

In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis?

Answer 1

Hi there!

I would argue that changing basis is most useful for the purpose of obtaining a diagonal matrix. This makes it a pretty powerful and important tool, because diagonal matrices are much much easier to deal with than general matrices. For example, taking the 100th power of even 10x10 diagonal matrix is easy (it is a diagonal matrix with diagonal elements exactly equal to the 100th power of the corresponding diagonal elements in the original matrix), but taking the 100th power of a 10x10 general matrix becomes a ridiculous problem to do by hand.

Nearly all finite matrices can be diagonalized through a change of basis, as stated in the real and complex spectral theorems. If you'd like to learn more about spectral theory, leave a comment!

To answer your second question, eigenvalues are indeed intimately linked to the diagonalized matrix. In particular, the diagonal matrix that is obtained by a particular change of basis has diagonal elements that are exactly comprised of each eigenvalue, including multiplicities. This becomes a bit complicated to explain in one text response, but I'm happy to elaborate or set up a short online session if you are interested.