Eigenvalues of n x n matrix

[coldstone]

Just wondering is there a way to get the characteristic equation of an n by n matrix without going through tedious calculations of solving multiple determinants of matrices?

[Quantumpencil]

I think you're basically stuck using the determinant, but there are many ways to simplify a matrix so that calculating it's determinant is easier.

[lurflurf]

There are many ways
-Hamiltan-Cayley
Find powers of the matrix and find the polynomial they satisfy
-eigenvalues
The terms of the characteristic polynomial are home geneous combinations of the eigenvalues
p=(x-eigenvalue1)(x-eigenvalue2)(x-eigenvalue3)...