Originally Posted by

**firebio** A=$\displaystyle \begin{bmatrix} 7 & 3 & 3 &2 \\ 0 & 1 & 2 & -4 \\ -8 &

-4& -5 &0 \\ -2 & 1 & 2 & 3\end{bmatrix}

$

Let T be the linear Operator in $\displaystyle R_n$ that has the given matrix A relative to the standard basis $\displaystyle E_n$.

Find the Spectral Decomposition of T.

I found that characteristic polynomial is $\displaystyle (x-3)^2 (x+1)(x-1)$

So $\displaystyle T=P_1-P_2+3P_3 $, where P are projections.

Trying to find E's such that$\displaystyle A=x_1E_1+x_2E_2+x_3E_3=E_1-E_2+3E_3 $

E are Projections matrix relative to $\displaystyle E_3$