Eigenvalues and Eigenvectors of diagonal marix

Question

Problem:

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.

Find $\sigma$($D$) and all eigenvectors of $D$.

My thoughts:

As the spectrum of $D$ is the set of all eigenvalues, then $\sigma$($D$) should be just $\lambda_1 \cdots \lambda_n$ = $\mathbb {\lambda_n}^{n}$ .

But how can I find the eigenvectors ? I know I have to calculate the $D$ - $\lambda I$.

Can someone help me?

Answer 1

If the matrix is diagonal, the eigenvectors are just the standard basis of $\mathbb{C}$:

$$ e_1 = (1, 0, \dots, 0)^t, \dots, e_n=(0,\dots, 0, 1)^t \ . $$