Let V Rn times n and let T elementof LV V be defined by TA

Let V = R^n times n and let T elementof L(V, V) be defined by T(A) = A - 1/n Tr(A) I. What is the minimal polynomial of T? Is T diagonalizable? What is the characteristic polynomial of T?

Solution

The nonzero monic polynomial in F[T] that kills A and has least degree is called the minimal polynomial of A in F[T].

for a matrix A M_n(F), viewed as an operator on Fⁿ , is that its minimal polynomial is the polynomial f(T) of least degree such that f(A) is the zero matrix

Hence the minimal polynomial here is the one that remove A and has the least degree.

(b) Here A is the linear transformation of finite order on a complex vector space

For T(A) =A - 1/nT_r(A)I

Here the minimal polynomial can be factorized Hence T is diagnolizable.

(c) f the characteristic polynomial is not the minimal polynomial then the minimal polynomial divides one of the quadratic factors

the minimal polynomial divides the characteristic polynomial, every root of the minimal polynomial (possibly in an extension of F) is an eigenvalue. The converse is also true: