# Thread: Linear Maps and polynomials

1. ## Linear Maps and polynomials

Given a linear map $f:V\longrightarrow V$, in the form of a square matrix A, I can calculate the characteristic polynimial $c\sb{f}(x)$ in x, which is det(A-xI).

How do I calculate the minimal polynomial $m\sb{f}(x)$? The minimal polymonial is defined as the unique polynomial with
(i) leading coefficient is 1 (for uniqueness)
(ii) it is the polynomial of least degree with $m\sb{f}(A) = 0$

2. Originally Posted by TweedyTL
Given a linear map $f:V\longrightarrow V$, in the form of a square matrix A, I can calculate the characteristic polynimial $c\sb{f}(x)$ in x, which is det(A-xI).

How do I calculate the minimal polynomial $m\sb{f}(x)$? The minimal polymonial is defined as the unique polynomial with
(i) leading coefficient is 1 (for uniqueness)
(ii) it is the polynomial of least degree with $m\sb{f}(A) = 0$

There's a basic theorem that states that the min. pol. of A has the very same irreducible factors as the char. pol., so you have to prove every possible combination of powers of the irred. factors of the char. pol.
For example, if the char. pol. is $x^3(x-1)^2(x-4)$, the min. pol. can be $x(x-1)(x-4)\,,\,\,x^2(x-1)(x-4)\,,\,\,x^3(x-1)(x-4)\,,\,\,x(x-1)^2(x-4)\,,\,etc.$, so you have to evaluate $A(A-I)(A-4I)\,,\,\,A^2(A-1)(A-4I)\,,\,etc.$ to check whats the pol. of minimal degree which A is a zero of.

Tonio