How to determine if two matrices have same eigenbasis

Question

I understand the process of finding eigenvalues and eigenvectors, but not so much the concept of an eigenbasis. Is it true in general that matrices that share a common set of eigenvectors can be described as having the same eigenbasis? For example, $A=A^{-1}$ I believe has the same set of eigenvectors but not eigenvalues - should this have the same eigenbasis?

Answer 1

I assume that you are using the language of Quantum Mechanics when you talk about eigenbasis. An eigenbasis is just a basis for your vector space s.t any vector in that basis is an eigenvector of some fixed operator that is acting on the vector space.

So in that sense, yes; if two matrices have the same eigenvector (not necessarily the same eigenvalues), they have the same eigenbasis.

Answer 2

An "eigenbasis" is simply a basis for the vector space that consists of eigenvectors of some linear transformation. A, (assuming such a basis exists). Yes, v is an eigenvector of A, corresponding to eigenvalue $\lambda$, if and only if $Av= \lambda v$. If A is invertible (in which case $\lambda$ is not 0) then, taking $A^{-1}$ of both sides, $v= \lambda A^{-1}v$ and, dividing both sides by $\lambda$, $\frac{1}{\lambda}v= A^{-1}v$. That is, the eigenvalues of $A^{-1}$ are the reciprocals of the eigenvalues of A with the same eigenvectors.

As for your question about an "eigenbasis", IF there exist a basis consisting entirely of eigenvectors (sometimes called a "complete set" of eigenvectors), then there exist an infinite number of such "eigenbases" so I don't know what you mean by "the same eigenbasis".