Eigenvectors/2 1 -2+i 0 i 1+i 0 0 -1+2i/Characteristic polynomial/Eigenvalues and eigenspaces/Diagonal matrix/Exercise

We consider the linear mapping

$\varphi \colon \mathbb {C} ^{3}\longrightarrow \mathbb {C} ^{3}$

that is given by the matrix

${}A={\begin{pmatrix}2&1&-2+{\mathrm {i} }\\0&{\mathrm {i} }&1+{\mathrm {i} }\\0&0&-1+2{\mathrm {i} }\end{pmatrix}}\,,$

with respect to the standard basis.

a) Determine the characteristic polynomial and the eigenvalues of ${}A$.

b) Compute, for every eigenvalue, an eigenvector.

c) Establish a matrix for ${}\varphi$ with respect to a basis of eigenvectors.

Create a solution