Calculating Eigen Values / Vectors of large matrices I want to know how to calculate the Eigenvalues / Eigenvectors of large matrices.
I am fairly confident that I can calculate the Eigen properties of matrices that of size $2x2$ but I'm confused on how to calculate Eigenvalues and Eigenvectors on matrices that are bigger, for example:
$$A = \begin{pmatrix}
1 & 2 &3  &4  &5  &6 \\ 
7 & 8 &9  &10  &11  &12 \\ 
13 &14  &15  &16  &17  &18 \\ 
19 &20  &21  &22  &23  &24 \\ 
25 &26  &27  &28  &29  &30 \\ 
31 & 32 & 33 &34  &35  &36 
\end{pmatrix}$$
Or any size matrix. I can calculate the Covariance matrix, and Determinant of any given matrix but cannot seem to figure out how I would go about calculating the Eigenvalues and Eigenvectors for larger matrices of $2x2$.
Can anyone offer any help OR suggest some light reading around this topic?
 A: The eigenvalues come as solutions of the characteristic equation, $\det(A-\lambda I) = 0$. For 5 or more variables (i.e., matrices with dimension 5 or larger), it is impossible to algebraically solve a polynomial in closed form (http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html), so you have to resort to numerical solutions in general.
For dimension 4 or lower, there are closed form solutions to every polynomial equation (higher-dimensional versions of the quadratic formula). For such matrices, you could use the explicit formulas on the characteristic equation to find the eigenvalues. But these formulas are ridiculously complicated (see http://upload.wikimedia.org/wikipedia/commons/9/99/Quartic_Formula.svg), so it is better to just use a computer.
A: There exist many such algorithms, already built-in in Matlab or Python (SciPy or NumPy). Such algorithms start from the QR algorithm and then modify it. Since these are numerical in nature, you may need to worry about stability/accuracy.
