Suppose RT R S1 TSF3 such that epsilon LF3 each have 2 6 7

Suppose R,T R = S^-1 TSF^3) such that epsilon L(F^3) each have 2, 6, 7 as eigenvalues. Prove that there exists an invertible operator S epsilon L(

Solution

As 3x3 matrix has 3 distinct eigen values, 2,6,7

we have got 3 distinct eigen vectors.

Hence If we construct a matrix S with eigen vectors as Columns

we find that S is non singular as the eigen vectors are distinct.

Hence S got an inverse

Let T be the diagonal matrix with entries in the diagonal as 2, 6,7

This proves that

S^-1TS = R