The Question: What could be, you think, the challenges in defining the characteristic polynomial for operators on real vector spaces?
No,there is no challenge in defining the characteristic polynomial for operator on real vector space.
Suppose , is a linear operator on a vector space V of finite dimension. We define the characteristic polynomial to be the characteristic polynomial of any matrtix representation of .
As we known that if are matrix representation of then 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 .
The Question: What could be, you think, the challenges in defining the characteristic polynomial for operators...
The only way I can think of is to show they have the same characteristic polynomial; thus they have the same eigenvalues, But the question asked not to use determinants. 4) Prone that if I is in Mann (R) matrix A end at here the same eigenvalues. (Do not use determinent) Here at means the transpose of A.
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