I am struggling with the following question:
When is it true that I can use the (right, eventually) eigenvectors of a finite dimensional matrix as a basis and thus write down an eigenvector expansion of a given vector?
I am sure about the fact that an answer to my question is the spectral theorem: for symmetric matrices, I can. Moreover, in this case the eigenvectors are orthogonal.
My problem is more with non-symmetric matrices. Let me explain that:
- First of all, a "logic" consideration: if we can always use the right eigenvectors of a non-symmetric matrix as a basis, I think it would be a well know result (in fact, the spectral theorem is well known).
- Looking on books by notable authors, I get confused. Everybody says "if the matrix is symmetric then the eigenvectors are a basis". However, I have never read a statement about non-symmetric matrices. Despite it, some authors just use the right eigenvectors as a basis. Some other, instead, they say that they assume they can use the right eigenvectors as a basis.
- Finally, the power method and the Jordan form: the power method is based on the fact that a generic vector can be written as a linear combination of eigenvectors of the matrix. It seems to me that the only condition required for this to be possible is for the Jordan form of the matrix to exist. On the other hand, it seems to me that the Jordan form of a matrix always exists. Thus I would conclude that I can always use the right eigenvectors of a non symmetric matrix. But it seems crazy to me that nobody ever says that.
Can somebody explain to me what is the state of the art about eigenvectors and their being a basis?