# Matrix Theory: An essential proof for eigenvector computations

I’ve avoided proofs unless absolutely necessary, but the relation between the same eigenvector expressed in two different bases, is important.
Given that AS is the linear transformation matrix in standard basis S, and AB is its counterpart in basis B, we can write the relation between them as:

$A_B = C^{-1}A_SC\\* A_S = CA_BC^{-1}$

where C is the similarity transformation. We’ve seen this relation already; check here if you’ve forgotten about it.