I'm learning about similar matrices and I've done problems such as calculating a diagonal matrix $D = C^{-1}AC$, similar to $A$. This involved finding eigenvectors and eigenvalues of $A$.
However, if I took any nonsingular matrix $S$ with compatible dimensions with $A$, wouldn't $S^{-1}AS$ be similar to $A$?
Aside from finding a similar diagonal matrix, I'm having some trouble understanding the role of eigenvectors and eigenvalues in similar matrices. Also, what's the significance of a diagonal matrix similar to some matrix $A$?
Best Answer
Note that if $$C^{-1}AC=D$$ where $D$ is diagonal, then $$ C^{-1}A^kC=D^k$$
that is $$A^k = CD^kC^{-1}$$
Since it is very straight forward to find powers of the diagonal matrix $D$ this process makes finding the powers of $A$ straight forward as well.
Same goes with exponential matrices which are use in solving systems of differential equations.
In terms of eigenvalues, note that a matrix and its similar matrices have the same eigenvalues but they do not necessarily share the same eigenvectors.