Question

The Question: What could be, you think, the challenges in defining the characteristic polynomial for operators on real vector spaces?

The characteristic polynomial is only defined in this section for operators on complex vector spaces. What do you think are the challenges in defining the characteristic polynomial for operators on real vector spaces?

Answer 1

No,there is no challenge in defining the characteristic polynomial for operator on real vector space.

Suppose ,
T:V →V
is a linear operator on a vector space V of finite dimension. We define the characteristic polynomial $\Delta (t) \ \text{of} \ T$ to be the characteristic polynomial of any matrtix representation of .

As we known that if $A \ \text{and} \ \ B$ are matrix representation of then
B=P-AP
where is a change of basis matrix.Thus and are similar.

But again we known that similar matrices have same characteristic polynomial.

Accordingly ,the characteristic polynomial of is independent of the particular basis in which the matrix representation of is computed.

So there is no problem in defining the characteristic polynomial for operator on real vector space.

Underlying field are important ,when we talking about diagonalization of a linear operator .