Characteristic polynomial of a matrix — algorithm

Question

Is there a classical algorithm to compute the coefficients of the characteristic polynomial of a real matrix, for small matrix sizes (say up to $10\times10$)?

Is there a specialized version for symmetric matrices?

Note that I am not after the roots of this polynomial (i.e., the eigenvalues).

Answer 1

Yes, Bocher's formula is exactlywhat you need. Let A be an nxn matrix and $ \beta_i $ the coefficient of $\lambda^{n-i} $ in $ det(\lambda I -A) $. Let $ t_i =tr(A^i) $.Then $$ \beta_1=-t_1 $$ $$ \beta_2=-(1/2)(\beta_1t_1+t_2) $$ $$ \beta_i=-(1/i)(\beta_{i-1}t_1+ ... +\beta_1t_{i-1}+t_i) $$ for $ i\le n $ This formula is valid with the entries in A in any commutative ring with identity in which the elements 2,3, ... ,n are all invertible.