Developing Intuition for Eigenvalues and Eigenvectors in Matrices

[EvLer]

I do not have a specific problem to show, but was wondering if someone could give tips on how to see or develop intuition on those eigenvectors for (2x2 and 3x3) matricies, i.e. which are the cases where they are obvious and how to see it, other than diagonal matrices (i.e. only diagonal is non-zero from upper left to lower right).

Thanks.

 

[HallsofIvy]

Well, a triangular matrix (one that has only zeroes below the main diagonal [or only zeroes above the main diagonal) also has its eigenvalues on the main diagonal. Other than that, solve the eigenvalue equation!

 

[AlephZero]

For a general matrix, there is no "intuition" about finding eigenvalues and eigenvectors.

In one sense, finding all the eigenpairs is the hardest question that can be asked about a general matrix. If you know all the eigenpairs, then you can easily transform the matrix into diagonal form, and any other question you can ask about it becomes trivial. In other words, the eigenpairs contain "all the information you can possibly know" about the matrix in an easy-to-use form - but there's no such thing as a free mathematical lunch!

 

[matt grime]

That's a very nice way of putting it AlephZero. Essentially, matrices are linear maps, and they're only ever determined up to conjugacy, and over C at least, this is completely determined by its Jordan Canonical form, which is precisely the information of its eigenvalues and dimensions of eigenspaces.

Of course, some intuition, or knowledge, is useful - a real symmetric matrix, or a hermitian matrix, has real eigenvalues. The sum of the eigenvalues is the trace, the product is the determinant (modulo some warnings about multiplicities).