# Eigenvalues and Eigenvectors of diagonal marix

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?

If the matrix is diagonal, the eigenvectors are just the standard basis of $$\mathbb{C}$$:
$$e_1 = (1, 0, \dots, 0)^t, \dots, e_n=(0,\dots, 0, 1)^t \ .$$