Question

Define transformation T: R^3 -> R^3 by T(x)=Ax where A (3x3) matrix and has eigenvalues 5 and -2. Is A similar to a diagonal matrix? explain

Answer 1

If it has 2 eigenvalues then one of them has algebraic multiplicity equal 2. Let
it be 5. Then
if matrix of transformation has form
5 1 0
0 5
0
0 0 -2
then it is its Jordan form, and it cannot be diagonalized.