Characteristic Polynomial - Characteristic Polynomial of A Product of Two Matrices

Characteristic Polynomial of A Product of Two Matrices

If A and B are two square n×n matrices then characteristic polynomials of AB and BA coincide:

More generally, if A is m×n-matrix and B is n×m matrices such that m<n, then AB is m×m and BA is n×n matrix. One has

To prove the first result, recognize that the equation to be proved, as a polynomial in t and in the entries of A and B is a universal polynomial identity. It therefore suffices to check it on an open set of parameter values in the complex numbers. The tuples (A,B,t) where A is an invertible complex n by n matrix, B is any complex n by n matrix, and t is any complex number from an open set in complex space of dimension 2n2 + 1. When A is non-singular our result follows from the fact that AB and BA are similar: