Characteristic polynomial

The characteristic polynomial isIts roots are precisely the eigenvalues of $A$.

For square matrices $A,B$ over a field,If $A$ is invertible, the products are similar matrices. In general, apply that case to $(A-sI)B$ and $B(A-sI)$ for all but finitely many $s$, then use polynomial identity in $s$.

Ifthe Faddeev–LeVerrier algorithm computes the coefficients recursively from traces of powers:

For real matrices $A,B$, $p(t)=\det (A+tB)$ is a real polynomial. If $p(i)\ne 0$, it is not the zero polynomial, so some real $\lambda$ satisfies $p(\lambda )\ne 0$.

Every square matrix over an algebraically closed field is similar to an upper-triangular matrix. Choose an eigenvector, extend it to a basis, and apply induction to the induced lower-right block.