Find the spectral decomposition of $A$ $$
A= \begin{pmatrix} -3 & 4\\ 4 & 3 
\end{pmatrix}
$$ 
So i am assuming that i must find the evalues and evectors of this matrix first, and that is exactly what i did.
The evalues are $5$ and $-5$, and the evectors are $(2,1)^T$ and $(1,-2)^T$
Now the spectral decomposition of $A$ is equal to $(Q^{-1})^\ast$ (diagonal matrix with corresponding eigenvalues) * Q
$Q$ is given by [evector1/||evector1|| , evector2/||evector2||]
and for Q i got the matrix 
$$
Q= \begin{pmatrix} 2/\sqrt{5} &1/\sqrt{5} \\ 1/\sqrt{5} & -2/\sqrt{5} 
\end{pmatrix}
$$ 
the inverse of Q is the matrix...
$$
 \begin{pmatrix} 2 \sqrt{5}/5 & \sqrt{5}/5 \\ \sqrt{5}/5 & -2 \sqrt{5}/5 
\end{pmatrix}
$$ 
and the diagonal matrix with corresponding evalues is
$$
A= \begin{pmatrix} 5 & 0\\ 0 & -5 
\end{pmatrix}
$$ 
so now i found the spectral decomposition of $A$, but i really need someone to check my work.
Did i take the proper steps to get the right answer, did i make a mistake somewhere?

I think of the spectral decomposition as writing $A$ as the sum of two matrices, each having rank 1.  Let $A$ be given.  Then compute the eigenvalues and eigenvectors of $A$.  Then 
$$
A = \lambda_1P_1 + \lambda_2P_2
$$
where $P_i$ is an orthogonal projection onto the space spanned by the $i-th$ eigenvector $v_i$.  
In your case, I get $v_1=[1,2]^T$ and $v_2=[-2, 1]$ from Matlab.  Then we have:
$$
\left[ \begin{array}{cc}
-3 & 4 \\
4 & 3\\
\end{array} \right] =
5\left[ \begin{array}{cc}
1/5 & 2/5 \\
2/5 & 4/5\\
\end{array} \right] -
5\left[ \begin{array}{cc}
4/5 & -2/5 \\
-2/5 & 1/5\\
\end{array} \right] 
$$
Each $P_i$ is calculated from $v_iv_i^T$.
