Suppose that A and B are nxn matrices and that the character

Suppose that A and B are nxn matrices and that the characteristic polynomial of A divides the characteristic polynomial of B. Prove that A and B have exactly the same eigenvalues.

Solution

Since, A and B are nxn matrices so their characterisitic polynomials have degree n

Let, f(t) ,g(t) denote characteristic polynomials of A and B respectively

f(t) and g(t) are of degree n

So, f(t) divides g(t) implies g(t) is a multiple of f(t) ie

g(t)=kf(t) where, k is some non zero constant

HEnce, f and g have the same roots and hence A and B have the same set of eigenvalues since roots of characteristic polynomial are the eigenvalues