# eigenvalues and eigenvectors of Diagonalisable matrices

Let $$D$$:= diag($$\lambda_1, \ldots, \lambda_n$$), i.e., $$D$$ is a diagonal matrix in $$\mathbb{C}^{n\times n}$$ with entries $$\lambda_1, \ldots, \lambda_n$$$$\mathbb{C}$$ on its diagonal.

Furter let $$U\in\mathbb{C}^{n\times n}$$ be an invertible matrix and define $$A$$ := $$U^{−1}$$DU$$, i.e. A$$ is similar to $$D$$. Find the eigenvalues and eigenvectors of $$A$$.

As $$U$$ is invertible and $$D$$ is diagonal I think that $$A$$ should also be a diagonal matrix.

And as $$A$$ is similar to $$D$$ then I suppose that $$A$$ has the same eigenvalues as $$D$$, namely $${\lambda n}^{n}$$.

Could someone tell me if I am on the right way?

## 1 Answer

If $$u_i$$ is an eigenvector of $$D$$: $$Du_i=\lambda_iu_i$$ than, from: $$A=U^{-1}DU\Rightarrow AU^{-1}=U^{-1}D$$

we have: $$A(U^{-1}u_i)=U^{-1}Du_i=U^{-1}\lambda_iu_i=\lambda_i(U^{-1}u_i)$$

that gives the eigenvectors for the same eigenvalues.

Geometrically you can think at $$U$$ as a change of basis, so $$A$$ represents the same linear transformation of $$D$$ in a new basis, so its eigenvectors are the same, but expressed in the new basis.