Question

What is role of geometric multiplicity in minimal polynomial?

Answer 1

Consider the Jordan matrices J1,J2J1,J2:

J1=⎛⎝⎜⎜⎜1000110000100011⎞⎠⎟⎟⎟  J2=⎛⎝⎜⎜⎜1000110001100001⎞⎠⎟⎟⎟J1=(1100010000110001)  J2=(1100011000100001)

both have only one eigenvalue, namely 1, so they both have algebraic multiplicity 4. They also both have geometric multiplicity 2, since there are 2 Jordan blocks in both matrices (check the Wikipedia article on Jordan normal formfor more information). However, they have different minimal polynomials:

mJ1(x)=(x−I)2mJ2(x)=(x−I)3mJ1(x)=(x−I)2mJ2(x)=(x−I)3

so the algebraic and geometric multiplicities do not determine the minimal polynomial.