# Linear algebra,spectral decomposition, minimal polynomial,diagonalizable

Consider the vector space $\mathbb C^4$ over the complex field . Let $T: V \to V$ be a linear operator defined by

$$T(z_1,z_2,z_3,z_4)=(0,z_2+\frac{\pi}{2}z_4,0,-\frac{\pi}{2}z_2+z_4)$$

a) Find the eigenvalues and the corresponding eigenspaces of $T$.

b) If $T$ is diagonalizable, find the projection operators $E_i$ such that $T$ has a spectral decomposition of the form $\lambda_1 E_1+\lambda_2 E_2+\cdots+\lambda_k E_k$.

c) Write the matrix representation of each projection operator $E_i$ w.r.t the standard basis for $V$.

d) Write the matrix of $T$ w.r.t the eigenvectors of $T$.

e) Write the minimal polynomial for $T$ and prove your answer.

Thank you :)

• Too many questions... what did you try yourself ? – Dietrich Burde May 20 '16 at 19:00

Suggestion: Start with representing the operator $T$ as a matrix with respect to the ordered basis $(e_2,e_4,e_1,e_3)$ where the vectors $(e_i)_{i=1}^4$ are the standard basis vectors of $\mathbb{C}^4$ (note that the order of the vectors in the basis is important!).