# Is there anyway to do this with eigen vectors and eigen basis?

So, say we have a square matrix A, and lets say that $$A^2, A^3....$$ so on are defined. Now If I want to find a general term $$A^n$$ is there any way to do that with eigen vectors. Take for example $$A = \begin{bmatrix} 2 & 0 \\ 1 & -1\end{bmatrix}$$. Like I would like to know if its possible to do it with eigen vectors instead of just trying to guess a pattern ? Like can we do something by converting the matrix into an eigen basis ?

• Putting a matrix to the n-th power is the very reason behind diagonalisation/trigonalization/jordanization of matrices. Nov 10, 2020 at 13:09

$$A = TDT^{-1},$$
where $$D$$ is a diagonal matrix such that its diagonal entries correspond to the eigenvalues of $$A$$, and the columns of $$T$$ are the eigenvectors of $$A$$.
$$A^n = \underbrace{TDT^{-1} \cdot TDT^{-1} \cdot \ldots \cdot TDT^{-1}}_{n~\text{times}}= \\ = T D (TT^-1) D (TT^-1) \ldots (TT^-1) D T^{-1}= \\ = T D I D I \ldots I D T^{-1} = \\ = T D^n T^{-1}.$$
Notice that $$D^n$$ is a diagonal matrix such that its diagonal entries are the $$n$$-th power of the eigenvalues of $$A$$.