I'm learning finding eigenvalues. I learned how to find simplistic eigenvalues for $3\times3$ matrix. By using below way. With this way I can only solve if I have simple determinant equation, like quadratic.
Find eigenvalues of $\lambda_i$, which are roots of $\det(A-\lambda I)=0$. $$\det(A-\lambda I)=\begin{vmatrix} 4 & 6 & 10 \\ 3 & 10 & 13 \\ -2 & -6 & -8\end{vmatrix} - \lambda \begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{vmatrix}$$
But for some of the matrices are big numbers and becoming cubic equation which won't have integer numbers as eigenvalues. I guess I need to simplify the matrix before somehow. Which I cannot find.
Can you please guide me how can I handle if matrices are not simple?
Any references or tutorials are appreciated.
I watched below video tutorial but author is using simple matrix in example: Eigenvalue and Eigenvector Computations Example
It's only at school that Eigenvalues are integer numbers ;-)
For the 3x3 case, you have the option of using the Cardano-Tartaglia formulas to solve the third degree equation. http://en.wikipedia.org/wiki/Cubic_function#Roots_of_a_cubic_function. They involve complex numbers.
Fourth degree can also be addressed by explicit formulas. http://en.wikipedia.org/wiki/Quartic_function#Solving_a_quartic_equation
As induced by Evariste Galois at the age of 17, there is no similar formula for degree 5 and above.
In practice, numerical methods are used. http://en.wikipedia.org/wiki/Eigenvalue_algorithm.
Also have a look at http://en.wikipedia.org/wiki/Eigenvalue_algorithm#3.C3.973_matrices.