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http://en.wikipedia.org/wiki/Characteristic_polynomial#Characteristic_equation

According to the above Wiki, the characteristic equation is very easy to solve if the matrix is a triangular matrix, or the $n$ is small. I want to know if there's any theorem that could help me find the eigenvalue/eigenvectors of an arbitrary matrix?

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If the matrix is triangular, it's not 'very easy': it's trivial. The eingenvalues are in the diagonal. –  leonbloy Jun 14 '11 at 15:34

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Well, suppose you have an arbitrary $n \times n$ matrix, the characteristic polynomial will be of degree $n$. Finding roots of an $n\:\text{th}$ degree polynomial is not so easy. You may want to see this wikipedia page:

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