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 $\mathbb{C}$:

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.